Inclusion Criteria:

          -  Patients with locally advanced or metastatic RMC histologically confirmed by expert
             pathology review and loss of SMARCB1 staining by immunohistochemistry. Also eligible
             are patients with other rare SMARCB1-negative tumors of the kidney, such as advanced
             or metastatic unclassified renal cell carcinoma with medullary phenotype (a rare RMC
             variant occurring in individuals without sickle hemoglobinopathies), and adult-onset
             malignant rhabdoid tumors. The principal investigator (PI) is the final arbiter in
             questions related to eligibility.

          -  Patients will be eligible regardless of whether they have had prior nephrectomy or
             still have their primary tumor in-situ. Patients with prior nephrectomy can be
             screened for enrollment at any time following the procedure.

          -  Patients must have at least one measurable site of disease, defined as a lesion that
             can be accurately measured in at least one dimension (longest diameter to be recorded)
             and measures >= 15 mm with conventional techniques or >= 10 mm with more sensitive
             techniques such as magnetic resonance imaging (MRI) or spiral computed tomography (CT)
             scan. If the patient has had previous radiation to the marker lesion(s), there must be
             evidence of progression since the radiation. Patients with disease confined to bone
             may be eligible if a measurable lytic defect is present or a serum marker is elevated
             (> 4 x upper limit of normal [ULN]). The PI is the final arbiter in questions related
             to measurability.

          -  Patients can have received prior immunotherapies such as anti-PD1, anti-PD-L1, or
             anti-CTLA-4 immune checkpoint inhibitors, or targeted therapies such as sunitinib,
             pazopanib, axitinib, cabozantinib, bevacizumab, erlotinib, and everolimus, or any
             cytotoxic chemotherapy regimens with the exception of regimens using a combination of
             gemcitabine >= 800 mg/m^2 plus adriamycin >= 30 mg/m^2. In addition, the total
             lifetime dose of doxorubicin prior to enrollment must not exceed 382 mg/m^2 as these
             would preclude patients from receiving at least 4 cycles of induction therapy of the
             trial protocol. Patients must not have received any proteasome inhibitors such as
             bortezomib or carfilzomib.

          -  There must be evidence of progression on or after last treatment regimen received.

          -  Eastern Cooperative Oncology Group (ECOG) performance status 0-2. NOTE: If subject is
             unable to walk due to paralysis, but is mobile in a wheelchair, subject is considered
             to be ambulatory for the purpose of assessing their performance status.

          -  Adolescent patients age 12 years and older are allowed with signed assent and parental
             consent according to institutional guidelines and requirements, as long as their body
             surface area (BSA) is >=1.2 given that this is the lower limit for which the
             independence between BSA and ixazomib exposure has been ascertained.

          -  Hemoglobin >= 9 g/dl (treatment allowed). May receive transfusion (within 14 days of
             the first dose of the study drugs).

          -  Absolute neutrophil count >= 1000/uL. Without growth factor support (filgrastim or
             pegfilgrastim) for at least 14 days (within 14 days of the first dose of the study
             drugs).

          -  Platelets >= 75,000/uL. Platelet transfusions to help patients meet eligibility
             criteria are not allowed within 3 days before study enrollment (within 14 days of the
             first dose of the study drugs).

          -  Total bilirubin =< 1.5 mg/dl. For patients with Gilbert's disease, total bilirubin
             should be =< 3 mg/dL (=< 51.3 umol/L) (within 14 days of the first dose of the study
             drugs).

          -  Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase [SGOT]) or
             alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase [SGPT]) =< 2.5 X
             institutional ULN, except in known hepatic metastasis, wherein may be =< 5 x ULN as
             per current American Society of Clinical Oncology recommendations. Approximately 15%
             of patients with RMC develop liver metastases (within 14 days of the first dose of the
             study drugs).

          -  Creatinine clearance > 30 mL/kg/1.73 m^2. If creatinine is not < 1.5 x ULN, then
             calculate by Cockcroft-Gault methods or local institutional standard (within 14 days
             of the first dose of the study drugs).

          -  International normalized ratio (INR) and partial thromboplastin time (PTT) =< 1.5 x
             ULN prior to study entry. Therapeutic anticoagulation with warfarin is allowed if
             target INR =< 3 on a stable dose of warfarin or other oral anticoagulant, or on a
             stable dose of low molecular weight (LMW) heparin for > 2 weeks (14 days) at the time
             of enrollment. Patients who have liver metastases resulting in INR and/or PTT > 1.5 x
             ULN and require chronic (>= 3 months) anticoagulation are not allowed.

          -  Patients must have a measured ejection fraction of at least 45% as measured by either
             multigated acquisition (MUGA) scan, echocardiogram, stress test, or ventriculography.

          -  Patients with controlled brain metastases are allowed on protocol if they have
             solitary brain metastasis that is asymptomatic and not requiring treatment or that was
             surgically resected or treated with radiosurgery or gamma knife, without recurrence or
             edema for 1 month (4 weeks).

          -  Female patients who

               -  Are postmenopausal for at least 1 year before the screening visit, OR

               -  Are surgically sterile, OR

               -  Women of childbearing potential (WOCBP) must have a negative serum or urine
                  pregnancy test (minimum sensitivity 25 IU/L or equivalent units of human
                  chorionic gonadotropin [HCG]) within 24 hours prior to the start of the study
                  drug.

          -  Women must not be breastfeeding.

          -  WOCBP must agree to practice 2 effective method(s) of contraception, at the same time,
             from the time of enrollment for the duration of treatment with study drug (s) plus 5
             half-lives of study drug (s) plus 30 days (duration of ovulatory cycle) for a total of
             5 months post treatment completion, or agree to practice true abstinence when this is
             in line with the preferred and usual lifestyle of the subject. (Periodic abstinence
             [e.g., calendar, ovulation, symptothermal, post-ovulation methods] and withdrawal are
             not acceptable methods of contraception). Complete abstinence is defined as complete
             avoidance of heterosexual intercourse and is an acceptable form of contraception for
             all study drugs.

          -  Abstinence is only acceptable when this is in line with the preferred and usual
             lifestyle of the subject. Periodic abstinence (e.g., calendar, ovulation,
             symptothermal, profession of abstinence for entry into a clinical trial,
             post-ovulation methods) and withdrawal are not acceptable methods of contraception.
             Subjects who choose complete abstinence are not required to use a second method of
             contraception, but female subjects must continue to have pregnancy tests. Acceptable
             alternate methods of highly effective contraception must be discussed in the event
             that the subject chooses to forego complete abstinence.

          -  Men who are sexually active with WOCBP, even if surgically sterilized (i.e., status
             post-vasectomy), must agree to follow instructions for method(s) of contraception for
             the duration of treatment with study drug (s) plus 5 half-lives of study drug (s) plus
             90 days duration of sperm turnover) for a total of 5 months post-treatment completion.

          -  Azoospermic males and WOCBP who are continuously not heterosexually active are exempt
             from contraceptive requirements. However WOCBP must still undergo pregnancy testing as
             described in these sections.

        Exclusion Criteria:

          -  Patients diagnosed or treated for another malignancy within 2 years before study
             enrollment or previously diagnosed with another malignancy and have any evidence of
             residual disease. Patients with nonmelanoma skin cancer or carcinoma in situ of any
             type are not excluded if they have undergone complete resection. Patients with another
             malignancy on watch and wait without any needing of treatment can be enrolled in this
             study.

          -  Patients must not have received anticancer therapies (including chemotherapy and
             targeted therapy) within 2 weeks (14 days) or 5 half-lives (whichever is shorter)
             prior to study day 1. Patients who have completed palliative radiation therapy more
             than 14 days prior to study day 1 are eligible. If the involved field is small, 7 days
             will be considered a sufficient interval between treatment and administration of the
             ixazomib.

          -  Patients, who have had a major surgery or significant traumatic injury (injury
             requiring > 4 weeks [28 days] to heal) within 4 weeks (28 days) of start of study
             drug, patients who have not recovered from the side effects of any major surgery
             (defined as requiring general anesthesia) or patients that are expected to require
             major surgery, other than cytoreductive nephrectomy +/- retroperitoneal lymph node
             dissection, during the course of the study.

          -  Known history of testing positive for human immunodeficiency virus (HIV) or known
             acquired immunodeficiency syndrome (AIDS).

          -  Positive test for hepatitis B virus (HBV) using HBV surface antigen (HBVsAg) test
             (regardless of HBV deoxyribonucleic acid [DNA] levels or IgM hepatitis B virus core
             antibody [anti-HBc] status) or positive test for hepatitis C virus (HCV) using HCV
             ribonucleic acid (RNA) or HCV antibody test indicating acute or chronic infection. If
             hepatitis C antibody test is positive then active infection has to be confirmed by
             hepatitis C RNA testing for the patient to be excluded.

          -  Patient has >= grade 3 peripheral neuropathy, or grade 2 peripheral neuropathy with
             pain on clinical examination during the screening period.

          -  Known gastrointestinal (GI) disease or GI procedure that could interfere with the oral
             absorption or tolerance of ixazomib, including difficulty swallowing, refractory
             nausea and vomiting, uncontrolled diarrhea, malabsorption, significant small bowel
             resection or gastric bypass surgery, use of feeding tubes. The PI is the final arbiter
             in questions related to eligibility.

          -  Systemic treatment, within 14 days before the first dose of ixazomib, with strong
             CYP3A inducers (rifampin, rifapentine, rifabutin, carbamazepine, phenytoin,
             phenobarbital), or use of St. John's wort.

          -  Patients receiving any concomitant systemic therapy for renal cell cancer are
             excluded.

          -  Participation in other clinical trials, including those with other investigational
             agents not included in this trial, within 30 days of the start of this trial and
             throughout the duration of this trial.

          -  Patients who have any severe and/or uncontrolled medical conditions or other
             conditions that could affect their participation in the study such as:

               -  Symptomatic congestive heart failure of New York heart Association class III or
                  IV.

               -  Unstable angina pectoris, symptomatic congestive heart failure, myocardial
                  infarction within 6 months of start of study drug, serious uncontrolled cardiac
                  arrhythmia or any other clinically significant cardiac disease.

               -  Severely impaired lung function as defined as oxygen (O2) saturation that is 92%
                  or less at rest on room air.

               -  Uncontrolled diabetes as defined by fasting serum glucose > 1 .5 x ULN.

               -  Systemic fungal, bacterial, viral, or other infection that is not controlled
                  (defined as exhibiting ongoing signs/symptoms related to the infection and
                  without improvement) despite appropriate antibiotics or other treatment.

               -  Known active or symptomatic viral hepatitis or chronic liver disease.
                  Uncontrolled adrenal insufficiency.

          -  Patients must not have history of other diseases, metabolic dysfunction, physical
             examination finding, or clinical laboratory finding giving reasonable suspicion of a
             disease or condition that contraindicates the use of ixazomib, gemcitabine, or
             doxorubicin, or that might affect the interpretation of the results of the study or
             render the subject at high risk from treatment complications.

          -  Uncontrolled brain or leptomeningeal metastases, including patients who continue to
             require glucocorticoids for brain or leptomeningeal metastases.

          -  Female patients who are pregnant or breast feeding, or adults of reproductive
             potential who are not using effective birth control methods as defined above. If
             barrier contraceptives are being used, these must be continued throughout the trial by
             both sexes. Hormonal contraceptives are not acceptable as a sole method of
             contraception.

          -  Failure to have fully recovered (i.e., =< grade 1 toxicity) from the reversible
             effects of prior chemotherapy.

          -  Known allergy to any of the study medications, their analogues, or excipients in the
             various formulations of any agent.