Inclusion Criteria:

          -  Patient must have a diagnosis of BRAF^V600E/K-mutated metastatic melanoma

          -  Patient must have had documented radiographic or clinical evidence of progressive
             disease while on combination BRAF/MEK inhibitor therapy. For Phase 2 only, no more
             than one intervening therapy since progression on BRAF/MEK inhibitor therapy is
             allowed. Subjects who have evidence of progression while on, or within 4 weeks of
             completing, combination BRAF/MEK inhibitor therapy in the adjuvant setting will be
             eligible

          -  PHASE 2 ONLY: Patient must have EZH2 alteration (somatic mutation or copy number
             alteration). Can be performed on either archival or fresh specimen. EZH2 alterations
             need to be documented by a Clinical Laboratory Improvement Act (CLIA)/Clinical
             Laboratory Improvement Program (CLIP)-certified next generation sequencing platform
             (Foundation One, Tempus, Guardant360, etc.)

          -  PHASE 2 ONLY: Patient must have measurable disease

          -  PHASE 2 ONLY: Patient must have at least one tumor lesion amenable to biopsy. If
             possible, this lesion should be different from the lesion used for following tumor
             measurements but is not required

          -  PHASE 2 ONLY: Patient must agree to planned pre-treatment and planned on-treatment
             biopsy. A pre-treatment biopsy will be optional if patient has an archival tissue
             block or 5 formalin-fixed paraffin-embedded (FFPE) slides available from specimen used
             to document presence of eligible EZH2 alteration that is deemed adequate for
             evaluation

          -  Patient must be >=18 years

               -  Because no dosing or adverse event data are currently available on the use of
                  tazemetostat in combination with Dabrafenib and Trametinib in patients < 18 years
                  of age, children are excluded from this study

          -  Patient must have an Eastern Cooperative Oncology Group (ECOG) performance status =< 2
             (Karnofsky >= 50%)

          -  Patients with symptomatic central nervous system (CNS) metastases are eligible if
             previously treated with surgery and/or radiation with no evidence of radiologic CNS
             recurrence or progression for 4 weeks and on a stable/tapering dose of steroid for at
             least one week prior to start of study drug. Patients with new or progressive
             asymptomatic CNS metastases are eligible

          -  Hemoglobin >= 9 g/dL

          -  Albumin >= 2.5 g/dL

          -  Leukocytes >= 3,000/mcL

          -  Absolute neutrophil count >= 1,500/mcL

          -  Platelets >= 100,000/mcL

          -  Serum bilirubin =< 1.5 x institutional upper limit of normal (ULN) except subjects
             with known Gilbert's syndrome

          -  Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase
             [SGOT])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase [SGPT])
             =< 2.5 x institutional ULN

          -  Creatinine =< 1.5 mg/dL OR calculated creatinine clearance (Cockcroft-Gault formula)
             >= 50 mL/min

          -  Glomerular filtration rate (GFR) >= 60 mL/min/1.73 m^2

          -  Patients with a prior (or concurrent, if enrolling in Phase 1) malignancy whose
             natural history or treatment does not have the potential to interfere with the safety
             or efficacy assessment of the investigational regimen are eligible for this trial

          -  Patient must be able to swallow and retain oral medication and must not have any
             clinically significant gastrointestinal abnormalities that may alter absorption such
             as malabsorption syndrome or major resection of the stomach or bowels

          -  Prothrombin time (PT)/international normalized ratio (INR) and partial thromboplastin
             time (PTT) =< 1.3 x institutional ULN. Prophylactic low dose warfarin may be given to
             maintain central catheter patency

          -  The effects of tazemetostat, and the combination of tazemetostat, dabrafenib and
             trametinib on the developing human fetus are unknown. Women of childbearing potential
             and all male patients must agree to the following:

               -  For women of childbearing potential: agreement to remain abstinent (refrain from
                  heterosexual intercourse) or use contraceptive methods that result in a failure
                  rate of < 1% per year during the treatment period, for 6 months after
                  tazemetostat discontinuation, or for 6 months after discontinuation of the
                  combination of tazemetostat, dabrafenib and trametinib. Should a woman become
                  pregnant or suspect she is pregnant while she is participating in this study, she
                  should inform her treating physician immediately

                    -  A woman is considered to be of childbearing potential if she is
                       postmenarcheal, has not reached a postmenopausal state (>= 12 continuous
                       months of amenorrhea with no identified cause other than menopause), and has
                       not undergone surgical sterilization (removal of ovaries and/or uterus)

                    -  Examples of contraceptive methods with a failure rate of < 1% per year
                       include bilateral tubal ligation, male sterilization, established, proper
                       use of hormonal contraceptives that inhibit ovulation, hormone-releasing
                       intrauterine devices, and copper intrauterine devices

                    -  Due to the potential of enzyme induction with tazemetostat, female subjects
                       who use hormonal contraceptives should use an additional barrier method of
                       birth control while on study treatment and for 6 months after
                       discontinuation of tazemetostat or the combination of tazemetostat,
                       dabrafenib and trametinib

                    -  The reliability of sexual abstinence should be evaluated in relation to the
                       duration of the clinical study and the preferred and usual lifestyle of the
                       patient. Periodic abstinence (e.g., calendar, ovulation, symptothermal, or
                       post ovulation methods) and withdrawal are not acceptable methods of
                       contraception

          -  Women of childbearing potential must have a negative urine or serum pregnancy test at
             screening

          -  For men: agreement to remain abstinent (refrain from heterosexual intercourse) or use
             contraceptive measures and agreement to refrain from donating sperm, as defined below:

               -  With female partners of childbearing potential or pregnant female partners, men
                  must remain abstinent or use a condom during the treatment period and for at
                  least 4 months after the last dose of study drug. Men must refrain from donating
                  sperm during this same period. In addition, female partners of male subjects
                  should adhere to the following:

                    -  Intrauterine device (IUD) (must provide medical documentation of IUD)

                    -  Hormonal contraceptive (partner must be on a stable dose of the same
                       hormonal contraceptive product for at least 4 weeks before receiving study
                       drug) AND a condom (hormonal contraceptives must be supplemented with
                       condoms)

               -  The reliability of sexual abstinence should be evaluated in relation to the
                  duration of the clinical study and the preferred and usual lifestyle of the
                  patient. Periodic abstinence and withdrawal are not acceptable methods of
                  contraception

          -  Ability to understand and the willingness to sign a written informed consent document.
             Participants with impaired decision-making capacity (IDMC) who have a
             legally-authorized representative (LAR) and/or family member available will also be
             eligible

          -  Have progressed on, been intolerant to / ineligible for, or refused prior standard of
             care anti-PD-1 based immunotherapy

        Exclusion Criteria:

          -  Previous therapy with a demethylating agent (i.e. decitabine) or previous therapy with
             an EZH2 inhibitor

          -  History of second malignancy not treated with curative intent

          -  History of life-threatening toxicity, including hypersensitivity, related to BRAF or
             MEK inhibitor therapy, or known immediate or delayed hypersensitivity reaction or
             idiosyncrasy to drugs chemically related to the study treatments, their excipients,
             and/or dimethyl sulfoxide (DMSO)

          -  Active infection requiring intravenous therapy

          -  Presence of untreated or progressive symptomatic CNS melanoma metastases. Diffuse
             leptomeningeal carcinomatosis or metastases causing spinal cord compression are
             exclusionary. Previously treated lesions should be stable for >= 4 weeks (must be
             documented by imaging). Subjects on a stable dose of corticosteroids for > 1 week can
             be enrolled. Subjects must also be off of enzyme-inducing anticonvulsants for > 4
             weeks

          -  Radiation therapy in the last 14 days. Palliative radiation to a localized area
             without residual toxicity requires a washout of at least 7 days

          -  Prior systemic anti-cancer therapy (chemotherapy, targeted therapy, immunotherapy,
             biologic therapy, or vaccine therapy) within the 3 weeks preceding the first dose of
             study treatment. For Phase 2 only, prior chemotherapy regimens are not permitted

          -  Use of other investigational drugs within 28 days (or five half-lives, whichever is
             shorter; with a minimum of 14 days from the last dose) preceding the first dose of
             study treatment and during the study

          -  Unresolved toxicity of National Cancer Institute Common Terminology Criteria for
             Adverse Events, version 5.0 (NCI CTCAE v5.0) grade 2 or higher from previous
             anti-cancer therapy, except alopecia, at the time of randomization

          -  Current use of a prohibited medication. Patients must not be treated with any
             medications or substances that are strong inhibitors or inducers of CYP3A or CYP2C8
             within 13 days prior to the first treatment through the end of the study. Current use
             of, or intended ongoing treatment with: herbal remedies (e.g., St. John's wort), or
             strong inhibitors or inducers of P-glycoprotein (Pgp) or breast cancer resistance
             protein 1 (Bcrp1) should also be excluded

          -  Human immunodeficiency virus (HIV)-positive patients on combination antiretroviral
             therapy are ineligible because of the potential for pharmacokinetic interactions with
             dabrafenib

          -  A history of hepatitis B Virus (HBV) or hepatitis C Virus (HCV) infection (with the
             exception of cleared HBV and HCV infection, which will be allowed)

          -  History of interstitial lung disease or pneumonitis

          -  Clinically significant bleeding diathesis or coagulopathy, including known platelet
             function disorders. Patients on anticoagulation with low molecular weight heparin or
             low dose warfarin are allowed

          -  History of myeloid malignancies, including myelodysplastic syndrome (MDS)

          -  Has abnormalities known to be associated with MDS (e.g. del 5q, chr 7 abn) and
             multiple primary neoplasms (MPN) (e.g. JAK2 V617F) observed in cytogenetic testing and
             DNA sequencing

          -  History of T-lymphoblastic lymphoma (T-LBL)/T-cell acute lymphoblastic leukemia
             (T-ALL)

          -  Patients with history of RAS mutation-positive tumors are not eligible regardless of
             interval from the current study. Note: Prospective RAS testing is not required.
             However, if the results of previous RAS testing are known, they must be used in
             assessing eligibility

          -  History or evidence of cardiovascular risks including any of the following:

               -  QT interval corrected for heart rate using Fridericia's formula (QT corrected by
                  Fridericia [QTcF]) >= 450 msec

               -  History of acute coronary syndromes (including myocardial infarction or unstable
                  angina), coronary angioplasty, or stenting within the past 24 weeks prior to
                  randomization

               -  History or evidence of current Class II, III, or IV heart failure as defined by
                  the New York Heart Association (NYHA) functional classification system

               -  Intra-cardiac defibrillators

               -  Abnormal cardiac valve morphology (>= grade 2) documented by echocardiogram
                  (ECHO); (subjects with grade 1 abnormalities [i.e., mild regurgitation/stenosis]
                  can be entered on study). Subjects with moderate valvular thickening should not
                  be entered on study

               -  History or evidence of current clinically significant uncontrolled cardiac
                  arrhythmias; clarification: Subjects with atrial fibrillation controlled for > 30
                  days prior to dosing are eligible

               -  Treatment refractory hypertension defined as a blood pressure of systolic > 140
                  mmHg and/or diastolic > 90 mm Hg which cannot be controlled by anti-hypertensive
                  therapy

               -  Left ventricular ejection fraction (LVEF) < institutional lower limit of normal
                  (LLN) by ECHO or multigated acquisition scan (MUGA)

               -  Known cardiac metastases

          -  Patients with uncontrolled intercurrent illness

          -  Patients with psychiatric illness/social situations that would limit compliance with
             study requirements

          -  Pregnant women are excluded from this study because of the potential for teratogenic
             or abortifacient effects. Because there is an unknown but potential risk for adverse
             events in nursing infants secondary to treatment of the mother with tazemetostat /
             dabrafenib / trametinib, breastfeeding should be discontinued prior to treatment

          -  Patients with uncontrolled diabetes

          -  Patients with a history of retinal vein occlusion (RVO), retinal pigment epithelial
             detachment (RPED) or other ophthalmologic toxicity