Inclusion Criteria:

          -  PHASE I PORTION ELIGIBILITY CRITERIA

          -  Patients must have BRAF^V600 mutant metastatic cancer irrespective of the histology or
             prior therapy; BRAF^V600 mutant status must be documented by a Clinical Laboratory
             Improvement Amendments (CLIA)-certified laboratory; use of an Food and Drug
             Administration (FDA)-approved test is preferred although other BRAF tests at a
             CLIA-certified laboratory may also be accepted

          -  Patients must have locally advanced unresectable stage IIIC or metastatic stage IV
             cancer with either progression to prior therapy or a newly diagnosed cancer that does
             not have an available treatment with curative intent

          -  Patients must have a complete physical examination and medical history within 28 days
             prior to registration

          -  Patients must have measurable or non-measurable disease; all measurable lesions must
             be assessed (by physical examination, computed tomography [CT], or magnetic resonance
             imaging [MRI] scan) within 28 days prior to registration; tests to assess
             non-measurable disease must be performed within 42 days prior to registration; all
             disease must be assessed and documented on the baseline tumor assessment form
             (Response Evaluation Criteria in Solid Tumors [RECIST] 1.1)

          -  All patients must undergo a CT or MRI of the brain within 42 days prior to
             registration; patients with asymptomatic brain metastases or previously treated brain
             metastases that are stable (i.e. not requiring corticosteroids) at the time of
             registration will be eligible

          -  Patients may have received prior systemic therapy (chemotherapy, immunotherapy,
             biologic therapy, or combination regimens); all adverse events associated with prior
             treatment must have resolved to < grade 1 prior to registration

          -  Patients progressing on a prior BRAF inhibitor-based therapy will be eligible, as are
             patients naive to BRAF inhibitor therapy; resistance to BRAF inhibitor-based therapy
             will be defined as progressive disease by RECIST 1.1 criteria while receiving therapy
             with a BRAF inhibitor (vemurafenib or dabrafenib, alone or in combination with a
             mitogen-activated protein kinase [MEK] inhibitor); this may be innate resistance
             (patients who never achieved a tumor response while on BRAF inhibitor therapy) or
             acquired resistance (progression after having a tumor response to BRAF inhibitor
             therapy); there will not be a period of break between progression on the prior BRAF
             inhibitor-based therapy and the start of dabrafenib, trametinib and GSK2141795

          -  Patients may have received prior surgery (for both the primary and stage IV disease);
             all adverse events associated with prior surgery must have resolved to =< grade 1
             prior to registration

          -  Patients may have received prior radiation therapy; all adverse events associated with
             prior radiation therapy must have resolved to =< grade 1 prior to registration

          -  Patients must be willing to submit blood for pharmacokinetics; sites must order S1221
             pharmacokinetic (PK) kit immediately after registration; the Southwest Oncology Group
             (SWOG) patient identification (ID) number must be provided on the S1221 PK Kit request
             form

          -  Patients must have available and be willing to submit baseline tissue taken at the
             time of disease progression to prior BRAF inhibitor-based therapy (either fresh frozen
             [preferred], or paraffin-embedded tumor blocks) OR must have a site of disease that
             can be biopsied within this study for translational medicine studies; tissue may be
             from an archival biopsy or a new biopsy after the patient has been registered to the
             protocol; since patients are referred to this protocol after progression on prior BRAF
             inhibitor-based therapy, the biopsy taken at the time of progression will be used as
             the baseline biopsy for this study; patients must be willing to submit plasma and
             whole blood for translational medicine studies

          -  Patients must have Zubrod performance status =< 1

          -  Absolute neutrophil count (ANC) >= 1,200/ul (obtained within 28 days prior to
             registration)

          -  Platelets >= 100,000/ul (obtained within 28 days prior to registration)

          -  Hemoglobin >= 9 g/dL (obtained within 28 days prior to registration)

          -  Total bilirubin =< 1.5 x institutional upper limit of normal (IULN) (or =< 2.5 x ULN
             for patients with Gilbert's syndrome) (with 28 days prior to registration)

          -  Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) =< 2.5 x IULN (or
             < 5 x IULN for patients with known liver metastases) (obtained within 28 days prior to
             registration)

          -  Serum albumin >= 2.5 g/dL (obtained within 28 days prior to registration)

          -  Serum creatinine =< 1.5 mg/dL OR measured or calculated creatinine clearance >= 50
             mL/min (obtained within 28 days prior to registration)

          -  Patient must have a left ventricular ejection fraction >= institutional lower limit of
             normal (LLN) by echocardiogram (ECHO) or multigated acquisition (MUGA) within 28 days
             prior to registration

          -  Patients with melanoma must have a serum lactate dehydrogenase (LDH) test performed
             within 28 days prior to registration

          -  Patients with human immunodeficiency virus (HIV) are eligible if they are not on
             antiviral agents and have adequate cluster of differentiation (CD)4 counts (>= 500
             mm^3)

          -  Patients receiving anticoagulation treatment are allowed to participate with
             international normalized ratio (INR) established within the therapeutic range

          -  Women of childbearing potential must have a negative pregnancy test within 14 days of
             registration

          -  Patients must be able to 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; patients who have
             feeding tubes can enroll in the study provided that the capsules do not need to be
             modified

          -  Patients or their legally authorized representative must be informed of the
             investigational nature of this study and must sign and give written informed consent
             in accordance with institutional and federal guidelines

          -  As a part of the Oncology Patient Enrollment Network (OPEN) registration process the
             treating institution's identity is provided in order to ensure that the current
             (within 365 days) date of institutional review board approval for this study has been
             entered in the system

          -  Patients must have a serum albumin >= 2.5 g/dL within 28 days prior to registration

          -  PHASE II PORTION ELIGIBILITY CRITERIA

          -  Patients must have histologically confirmed melanoma with BRAF^V600 mutation; patients
             must have stage IIIC or stage IV disease

          -  Patients must have received prior BRAF inhibitor therapy (e.g. dabrafenib,
             vemurafenib) within 56 days prior to registration; prior trametinib therapy is
             permitted; patients are not required to interrupt BRAF or MEK inhibitor therapy prior
             to the initiation of three agent combination therapy on study

          -  Patients must have measurable disease; all measurable lesions must be assessed (by
             physical examination, CT, or MRI scan) within 28 days prior to registration; tests to
             assess non-measurable disease must be performed within 42 days prior to registration;
             patients whose only measurable disease is within a previous radiation therapy port
             must demonstrate clearly progressive disease (in the opinion of the treating
             investigator) prior to registration; all disease must be assessed and documented on
             the baseline tumor assessment form (RECIST 1.1)

          -  Patients must have Zubrod performance status =< 2

          -  No other prior malignancy is allowed except for the following: adequately treated
             basal cell or squamous cell skin cancer, in situ cervical cancer, adequately treated
             stage I or II cancer from which the patient is currently in complete remission, or any
             other cancer from which the patient has been disease free for five years; 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

        Exclusion Criteria:

          -  Patients must not have a corrected QT (QTc) interval >= 480 msecs within 28 days prior
             to registration

          -  At the time of registration, patients must not be receiving any medications or
             substances that are strong inhibitors or inducers of cytochrome P450, family 3,
             subfamily A (CYP3A) or cytochrome P450, family 2, subfamily C, polypeptide 8 (CYP2C8);
             patients must not be planning to use herbal remedies (e.g., St. John's wort), or
             strong inhibitors or inducers of P-glycoprotein (Pgp) or breast cancer resistance
             protein 1 (Bcrp1)

          -  Patients must not have a history of acute coronary syndromes (including unstable
             angina), myocardial infarction within 6 months, coronary angioplasty, or stenting
             within the past 24 weeks; class II, III, or IV heart failure as defined by the New
             York Heart Association (NYHA) functional classification system; or history of known
             cardiac arrhythmias (such as atrial fibrillation) unless it has been stably controlled
             for > 30 days prior to registration; abnormal cardiac valve morphology (>= grade 2)
             documented by echocardiogram (subjects with grade 1 abnormalities [i.e., mild
             regurgitation/stenosis]) can be entered on study; subjects with moderate valvular
             thickening are not eligible

          -  Patients must not be pregnant or nursing due to unknown teratogenic side effects;
             women/men of reproductive potential must have agreed to use an effective contraceptive
             method; a woman is considered to be of "reproductive potential" if she has had menses
             at any time in the preceding 12 consecutive months; in addition to routine
             contraceptive methods, "effective contraception" also includes heterosexual celibacy
             and surgery intended to prevent pregnancy (or with a side-effect of pregnancy
             prevention) defined as a hysterectomy, bilateral oophorectomy or bilateral tubal
             ligation; hormonal contraception is not allowed due to drug interactions which can
             render hormonal contraceptives ineffective; however, if at any point a previously
             celibate patient chooses to become heterosexually active during the time period for
             use of contraceptive measures outlined in the protocol, he/she is responsible for
             beginning contraceptive measures

          -  Patient must not have uncontrolled intercurrent illness including, but not limited to,
             ongoing or active infection, previously diagnosed type 1 diabetes mellitus/type 2
             diabetes, psychiatric illness/social situations, symptomatic congestive heart failure,
             unstable angina pectoris, cardiac arrhythmia, that would limit compliance with study
             requirements; patients must not have any evidence of mucosal or internal bleeding;
             patients must not have a history of pneumonitis or interstitial lung disease; patients
             must not have received any major surgery within four weeks prior to registration

          -  Patients must not have an active hepatitis B virus (HBV) or hepatitis C virus (HCV)
             infection

          -  Patients must not have a history of allergic reactions attributed to compounds of
             similar chemical or biologic composition to dabrafenib or other agents used in this
             study including dimethyl sulfoxide (DMSO)

          -  Patients with known history or current evidence of retinal vein occlusion (RVO) are
             not eligible:

               -  History of RVO, or predisposing factors to RVO (e.g. uncontrolled glaucoma or
                  ocular hypertension, uncontrolled systemic disease such as hypertension, diabetes
                  mellitus, or history of hyperviscosity or hypercoagulability syndromes)

               -  Visible retinal pathology as assessed by ophthalmic exam that is considered a
                  risk factor for RVO such as:

                    -  Evidence of new optic disc cupping

                    -  Evidence of new visual field defects

                    -  Intraocular pressure > 21 mmHg

                    -  NOTE: Ophthalmic exam is required for all patients; exam must be obtained
                       within 28 days prior to registration

          -  Patients must not have uncontrolled hypertension (defined as systolic blood pressure >
             140 mm Hg and/or diastolic blood pressure > 90 mm Hg which cannot be controlled by
             anti-hypertensive therapy)