Ixazomib Citrate With Gemcitabine Hydrochloride and Doxorubicin Hydrochloride in Treating Patients With Urothelial Cancer That is Metastatic or Cannot Be Removed by Surgery

Inclusion Criteria: * All patients must have histologic demonstration of metastatic or locally unresectable transitional cell carcinoma of the urothelium; variant histology is allowed as long as there is an urothelial component present; the principal investigator (PI), will serve as the final arbiter of eligibility * All patients must have measurable or evaluable disease; in general, liver and lung lesions should be at least 1 cm, and patients with node-only disease should have lesions of \>= 1.5 cm in greatest dimension; 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 principal investigator is the final arbiter in questions related to measurability; patients with a three-dimensional mass or pelvic sidewall fixation on bladder examination under anesthesia are considered to have measurable disease * Patients must have had at least one prior therapy to be eligible for either phase I or II, unless they are either not candidates for or refuse cisplatin-based therapy * Phase I: patients are eligible with any number of prior regimens regardless of what those regimens contained (i.e. prior bortezomib or combination gemcitabine and adriamycin is acceptable) * Phase II: patients are eligible if their previous chemotherapy regimen did not contain bortezomib, carfilzomib, or other known proteasome inhibitor or a combination of gemcitabine \>= 800 mg/m\^2 plus adriamycin \>= 30 mg/m\^2; patients who receive sequential or alternating therapy as part of front-line treatment will be counted as having one prior regimen; patients who have failed prior neoadjuvant chemotherapy will be eligible for this trial * If prior history of ischemic heart disease or exposure to 200 mg/m\^2 of doxorubicin, patients must have a measured ejection fraction (either by multigated acquisition scan \[MUGA\], echocardiogram \[ECHO\], stress test, or ventriculography) of at least 45% * Alanine aminotransferase (ALT) and aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\]) levels =\< 3 x the upper limit of normal * Eastern Cooperative Oncology Group (ECOG) performance status (PS) =\< 2; patients with a PS of 3 are eligible if the performance status is due to their malignancy, and not a co-morbid medical condition (example \[ex\]: perineal pain impacting their ability to sit or ambulate, etc.) * Patients who: * Are postmenopausal for at least 1 year before the screening visit, OR * Are surgically sterile, OR * If they are of childbearing potential, agree to practice 2 effective methods of contraception, at the same time, from the time of signing the informed consent form through 90 days after the last dose of study drug, OR * Agree to practice true abstinence when this is in line with the preferred and usual lifestyle of the subject; (periodic abstinence \[eg, calendar, ovulation, symptothermal, post-ovulation methods\] and withdrawal are not acceptable methods of contraception) * Male patients, even if surgically sterilized (ie, status post-vasectomy), must agree to one of the following: * Agree to practice effective barrier contraception during the entire study treatment period and through 90 days after the last dose of study drug, OR * Agree to practice true abstinence when this is in line with the preferred and usual lifestyle of the subject; (periodic abstinence \[eg, calendar, ovulation, symptothermal, post-ovulation methods\] and withdrawal are not acceptable methods of contraception) * Voluntary written consent must be given before performance of any study related procedure not part of standard medical care, with the understanding that consent may be withdrawn by the patient at any time without prejudice to future medical care Exclusion Criteria: * Platelet count of \< 100 x 10\^9/L; platelet transfusions to help patients meet eligibility criteria are not allowed within 3 days before study enrollment * An absolute neutrophil count of \< 1.0 x 10\^9/L * A calculated creatinine clearance of \< 30 mL/min using Cockcroft Gault or measured by 24 hour urine * Patient has \>= grade 3 peripheral neuropathy, or grade 2 with pain on clinical examination during the screening period * Total bilirubin \>= 1.5 x the upper limit of the normal range (ULN) * Known allergy to any of the study medications, their analogues, or excipients in the various formulations of any agent * Female subject is pregnant or breast-feeding * Confirmation that the subject is not pregnant must be established by a negative serum beta-human chorionic gonadotropin (beta-hCG) pregnancy test (unless there is reasonable certainty that beta-hCG is coming from the tumor); pregnancy testing is not required for post-menopausal or surgically sterilized women * Participation in other clinical trials with investigational agents not included in this trial, within 30 days of the start of this trial and throughout the duration of this trial * Patients with significant atherosclerotic disease, as defined by: * Myocardial infarction within 6 months prior to enrollment or has New York Heart Association (NYHA) class III or IV heart failure uncontrolled angina, severe uncontrolled ventricular arrhythmias, or electrocardiographic evidence of acute ischemia or active conduction system abnormalities * Symptomatic congestive heart failure * Claudication limiting activity and * History of cerebrovascular events within the last year (including transient ischemic attack \[TIA\]) * Unstable angina * Patients with known active brain metastases; (subjects with previously treated brain metastases are eligible provided they are stable \[defined as without evidence of progression by imaging for at least four weeks prior to the first dose of trial treatment\] and neurologic symptoms have returned to baseline) * Radiotherapy within 28 days before enrollment; if the involved field is small, 14 days will be considered a sufficient interval between treatment and administration of the therapy * 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 pathologically confirmed completely resected prostate cancer no higher than stage pT2a and no biochemical relapse, or pT2c tumors involving less than 5% of the prostate and no biochemical relapse, nonmelanoma skin cancer or carcinoma in situ of any type are not excluded if they have undergone complete resection * Systemic treatment, within 14 days before the first dose of ixazomib, with strong inhibitors of cytochrome P450, family 1, subfamily A, polypeptide 2 (CYP1A2) (fluvoxamine, enoxacin, ciprofloxacin), strong inhibitors of cytochrome P450, family 3, subfamily A (CYP3A) (clarithromycin, telithromycin, itraconazole, voriconazole, ketoconazole, nefazodone, posaconazole) or strong CYP3A inducers (rifampin, rifapentine, rifabutin, carbamazepine, phenytoin, phenobarbital), or use of Ginkgo biloba or St. John's wort * Failure to have fully recovered (ie, =\< grade 1 toxicity) from the reversible effects of prior chemotherapy * Major surgery within 14 days before enrollment; the PI will serve as the final arbiter as to what constitutes major surgery * Infection requiring current intravenous antibiotic therapy; the PI will serve as the final arbiter regarding eligibility * Ongoing or active systemic infection, known active hepatitis B or C virus infection, or known human immunodeficiency virus (HIV) positive * Any serious medical or psychiatric illness that could, in the investigator's opinion, potentially interfere with the completion of treatment according to this protocol * Known gastrointestinal (GI) disease or GI procedure that could interfere with the oral absorption or tolerance of ixazomib including difficulty swallowing; patients who have had cystectomy with conduit, neobladder, or pouch using a portion of their terminal ileum are allowed if, the patient is stable without clinically significant metabolic disturbances OR stoma- and/or anastomosis-related complications OR post-surgical gut abnormalities that would compromise drug absorption