Combination Chemotherapy and Donor Stem Cell Transplant Followed by Ixazomib Citrate Maintenance Therapy in Treating Patients With Relapsed High-Risk Multiple Myeloma

Inclusion Criteria: * Relapsed multiple myeloma in patients that have been treated previously with autologous hematopoietic stem cell transplantation (auto-HCT), bortezomib and an immunomodulatory agent, AND with at least one of the following high-risk criteria * High-risk multiple myeloma defined by cytogenetic or fluorescence in situ hybridization (FISH) detection of any one or more of the following: * Deletion 17p * Translocation t(4;14) * Translocation t(14;16) * Translocation t(14;20) * Chromosome 1q gain * Chromosome 1p deletion * Deletion 13q by conventional karyotyping (FISH only not acceptable) * Hypodiploidy * High-risk gene expression profiling (GEP) at the time of relapse * Beta-2 (B2) microglobulin \> 5.5 mg * Plasmablastic morphology (\> 2%) * Relapsed plasma cell leukemia * Chemo-sensitive disease; patients with relapsed plasma cell leukemia may have received systemic therapy including an autologous transplant but it is not required; patients with relapsed multiple myeloma (MM) must have received prior systemic therapy including an autologous transplant; patient must be in at least a PR at the time of transplant; early relapse from complete response will be allowed * Measurable disease at the time of relapse, defined as a monoclonal immunoglobulin spike on serum electrophoresis of \>= 1 gm/dL (immunoglobulin \[IG\]G) or \>= 0.5 gm/dL (IGA) and/or urine monoclonal immunoglobulin spike of \> 200 mg/24 hours and/ or involved free light chain (FLC) level \>= 10 mg/dl and the serum FLC ratio is abnormal * Non-secretors must have measurable disease such as plasmacytomas, or positron emission tomography (PET) avid lytic lesions or bone marrow plasmacytosis \>= 30% at the time of relapse to be eligible * The patient must have an available sibling or matched unrelated donor with at least a 7/8 human leukocyte antigen (HLA) match * Creatinine =\< 2.0 mg/dL * Ejection fraction \>= 45% * Diffusing capacity of the lungs for carbon monoxide (DLCO) \>= 50% * Forced expiratory volume in 1 second (FEV1)/forced vital capacity (FVC) \>= 50% predicted * Both men and women and members of all races and ethnic groups will be included * Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 (Karnofsky \>= 50%) * Willing to use adequate contraception for the duration of time on the study and for 90 days after the last therapy * Female patients must meet one of the following: * Postmenopausal for at least 1 year before the screening visit, OR * 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, AND * Must also adhere to the guidelines of any treatment-specific pregnancy prevention program, if applicable, 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) * Male patients, even if surgically sterilized (i.e., 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 * Must also adhere to the guidelines of any treatment-specific pregnancy prevention program, if applicable, 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 * 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 * DONOR: HLA genotypically identical sibling matched relative * DONOR: HLA matched unrelated donor according to Standard Practice HLA matching criteria: * Matched HLA-A, -B, -C, and -DRB1 alleles by high resolution typing * Only a single allele disparity will be allowed for HLA-A, B, or C as defined by high resolution typing Exclusion Criteria: * Previous allogeneic stem cell transplant * POEMS syndrome (plasma cell dyscrasia with polyneuropathy, organomegaly, endocrinopathy, monoclonal protein \[protein\] and skin changes) * Bilirubin \> 1.5 x the upper limit of normal * Alanine aminotransferase (ALT) and aspartate aminotransferase (AST) \> 2.5 x the upper limit of normal * Patients with \>= grade III or grade II with pain peripheral neuropathy (National Cancer Institute \[NCI\] Common Terminology Criteria for Adverse Events \[CTCAE\] version \[v.\] 4.03 criteria) * Receiving steroids \> the equivalent of 10 mg prednisone daily for other medical conditions, e.g., asthma, systemic lupus erythematosus, rheumatoid arthritis * Infection requiring systemic antibiotic therapy or other serious infection within 14 days before study enrollment * Ongoing or active systemic infection, active hepatitis B or C virus infection, or known human immunodeficiency virus (HIV) positive * 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 * Second malignancy requiring concurrent treatment or those with non-hematological malignancies (except non-melanoma skin cancers); cancer treated with curative intent \< 5 years previously will not be allowed unless approved by the protocol chair; cancer treated with curative intent \> 5 years previously is allowed * Other serious medical or psychiatric illness that could 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 * Radiotherapy within 14 days before enrollment; if the involved field is limited to a single site, 7 days will be considered a sufficient interval between treatment and administration of the ixazomib * Systemic treatment, within 14 days before the first dose of ixazomib, with strong inhibitors of 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 * Known gastrointestinal (GI) disease or GI procedure that could interfere with the oral absorption or tolerance of ixazomib including difficulty swallowing * Female patients who are lactating or have a positive serum pregnancy test during the screening period * Failure to have fully recovered (i.e., =\< grade 1 toxicity) from the reversible effects of prior chemotherapy * Major surgery within 14 days before enrollment * Central nervous system involvement * Participation in other clinical treatment trials, including those with other investigational agents not included in this trial, within 21 days of the start of this trial and throughout the duration of this trial * DONOR: Identical twin * DONOR: Donors unwilling to donate PBSC * DONOR: Pregnancy * DONOR: Infection with HIV * DONOR: Inability to achieve adequate venous access * DONOR: Known allergy to filgrastim (G-CSF) * DONOR: Current serious systemic illness * DONOR: Failure to meet institutional criteria for stem cell donation * DONOR: Patient and donor pairs must not be homozygous at mismatched allele