Ixazomib Citrate, Pomalidomide, Dexamethasone, Stem Cell Transplant in Treating Relapsed or Refractory Multiple Myeloma

Inclusion Criteria: * Previously treated ASCT naive MM patients, currently with relapsed or refractory disease who are being considered for single ASCT for relapsed disease; patients must be eligible to undergo a stem cell transplant as per institutional criteria for selection at the time of registration * Calculated creatinine clearance (using Cockcroft-Gault equation) \>= 30 mL/min * Absolute neutrophil count (ANC) \>= 1000/mm\^3 * Platelet count \>= 75 x 10\^9/L unless the participant has \>= 50% bone marrow infiltration in which case a platelet count of \>= 50 x 10\^9/L is allowed * Hemoglobin \>= 9.0 g/dL * Total bilirubin =\< 1.5 x the upper limit of the normal range (ULN) or if total bilirubin is \> 1.5 x ULN, the direct bilirubin must be =\< 2.0 mg/dL * Alanine aminotransferase (ALT) and aspartate aminotransferase (AST) =\< 3 x ULN or \< 5 x ULN if liver involvement * Patients with measurable disease defined as at least one of the following: * Serum monoclonal protein \>= 1.0 g/dL by protein electrophoresis * \>= 200 mg of monoclonal protein in the urine on 24-hour electrophoresis * Serum immunoglobulin free light chain \>= 10 mg/dL AND abnormal serum immunoglobulin kappa to lambda free light chain ratio * Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0, 1 or 2 * Willing to provide informed written consent * Negative pregnancy test done =\< 7 days prior to registration, for persons of childbearing potential only * Willing to follow strict birth control measures * Persons of childbearing potential, agree to one of the following: * 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) * Persons able to father a child: 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) * Willing to follow the requirements of the Pomalyst REMS program * Willing to return to return to enrolling institution for follow-up (during the Active Monitoring Phase of the study) * Willing to provide bone marrow samples under Institutional Review Board (IRB)#521-93 for correlative research purposes Exclusion Criteria: * Diagnosed or treated for another malignancy =\< 2 years prior to registration or previously diagnosed with another malignancy and have any evidence of residual disease * NOTE: If there is a history or prior malignancy, patient must not be receiving other specific treatment for their cancer; patients with non-melanoma skin cancer or carcinoma in situ of any type, or low-risk prostate cancer after curative therapy, are not excluded if they have undergone complete resection * NOTE: Platelet transfusions to help patients meet eligibility criteria are not allowed =\< 3 days prior to study registration * Any of the following: * Pregnant persons * Nursing persons * Persons of childbearing potential who are unwilling to employ adequate contraception * Co-morbid systemic illnesses or other severe concurrent disease which, in the judgment of the investigator, would make the patient inappropriate for entry into this study or interfere significantly with the proper assessment of safety and toxicity of the prescribed regimens; NOTE: this includes uncompensated heart or lung disease * Receiving any other investigational agent which would be considered as a treatment for the primary neoplasm; NOTE: Bisphosphonates are considered to be supportive care rather than therapy and are allowed while on protocol treatment * Patient has \>= grade 2 peripheral neuropathy, or grade 1 with pain on clinical examination during the screening period * Major surgery =\<14 days prior to registration * Systemic treatment with strong CYP3A inducers (rifampin, rifapentine, rifabutin, carbamazepine, phenytoin, phenobarbital) or use of Ginkgo biloba or St. John?s wort =\< 14 days prior to registration * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements. NOTE: Any electrocardiogram (ECG) abnormality at screening must be documented by the investigator as not medically relevant * Corrected QT (QTc) \> 470 milliseconds (msec) on a 12-lead ECG obtained during the screening period; Note: If a machine reading is above this value, the ECG should be reviewed by a qualified reader and confirmed on a subsequent ECG * Known human immunodeficiency virus (HIV) positive * Known hepatitis B surface antigen-positive status, or known or suspected active hepatitis C infection * Known allergy to any of the study medications, their analogues or excipients in the various formulations * Known gastrointestinal (GI) disease or GI procedure that could interfere with the oral absorption or tolerance of study treatment including difficulty swallowing * Diarrhea \> grade 1, based on the National Cancer Institute (NCI) Common Terminology Criteria for Adverse Events (CTCAE) version (v) 4.0 grading, in the absence of antidiarrheals * Failure to have fully recovered (i.e., =\< grade 1 toxicity) from the reversible effects of prior chemotherapy * Radiotherapy =\< 14 days prior to registration; NOTE: If the involved field is small, 7 days will be considered a sufficient interval between treatment and administration of the ixazomib * Central nervous system involvement with disease under study (myeloma), or concurrent AL amyloidosis or plasma cell leukemia * Patients that have previously been treated with ixazomib, or participated in a study with ixazomib whether treated with ixazomib or not * Prior stem cell transplantation for myeloma