Ixazomib Citrate in Treating Patients With Chronic Graft-versus-Host Disease

Inclusion Criteria: * 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of any study related procedure not part of standard medical care, with the understanding that consent may be withdrawn by the patient at any time without prejudice to future medical care * Female patients who: * Are postmenopausal for at least 1 year before the screening visit,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 or exclusively non-heterosexual activity 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 two effective contraception measures during the entire study treatment period and through 90 days after the last dose of study drug, OR * Agree to practice true abstinence or exclusively non-heterosexual activity 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) * Patients must have a diagnosis of a chronic GVHD according to the National Institute of Health (NIH) Consensus Criteria * Patients must have failed at least one prior line of systemic immune suppressive therapy for management of chronic GVHD * Absolute neutrophil count (ANC) \>= 1,000/mm\^3 * Platelet count \>= 75,000/mm\^3; platelet transfusions are not allowed within 3 days before study enrollment * Total bilirubin =\< 1.5 x the upper limit of the normal range (ULN) * Alanine aminotransferase (ALT) and aspartate aminotransferase (AST) =\< 3 x ULN * Calculated creatinine clearance \>= 30 mL/min Exclusion Criteria: * Female patients who are lactating or have a positive serum pregnancy test during the screening period * Major surgery within 14 days before enrollment * Does not include placement of venous access device, bone marrow biopsy, GVHD diagnostic biopsy, or other routine procedures in chronic GVHD or post-transplantation care * Uncontrolled infection within 14 days before study enrollment * Infection treated with appropriate antimicrobial therapy and without signs of progression/treatment failure does not constitute an exclusion criterion * Evidence of current uncontrolled cardiovascular conditions, including uncontrolled hypertension, uncontrolled cardiac arrhythmias, symptomatic congestive heart failure, unstable angina, or myocardial infarction within the past 6 months * Chronic hypertension on medical therapy does not constitute an exclusion criterion * Systemic treatment, within 14 days before the first dose of ixazomib, with strong inhibitors of CYP1A2 (fluvoxamine, enoxacin, ciprofloxacin), strong inhibitors of cytochrome 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 * 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 allergy to any of the study medications, their analogues, or excipients in the various formulations of any agent * Non-hematologic malignancy within the past 2 years with the exception of: * Adequately treated basal cell carcinoma, squamous cell skin cancer, or thyroid cancer * Carcinoma in situ of the cervix or breast * Prostate cancer of Gleason grade 6 or less with stable prostate-specific antigen levels * Cancer considered cured by surgical resection or unlikely to impact survival during the duration of the study * Patient has \>= grade 3 peripheral neuropathy, or grade 2 with pain on clinical examination during the screening period * Treatment with non-Food and Drug Administration (FDA) approved drug within 21 days of start of this trial * New systemic immune suppressive agent added for the treatment of chronic GVHD within 2 weeks prior to enrollment * Addition of a new systemic immune suppressive treatment simultaneously with ixazomib is also prohibited * Evidence of recurrent or progressive underlying malignant disease * Karnofsky performance status \< 70% * Life expectancy less than 6 months