Pomalidomide, Ixazomib Citrate, and Dexamethasone in Treating Patients With Previously Treated Multiple Myeloma or Plasma Cell Leukemia

Inclusion Criteria: * Previously treated myeloma, currently with extramedullary disease (defined as plasmacytoma outside bone marrow that is not contiguous with a bone lesion) with at least one lesion that has a single diameter of \>= 2 cm or plasma cell leukemia (defined as circulating plasma cells exceeding 5% of peripheral blood leukocytes or 0.5 X 10\^9/L or 200 cells/150000 events by flow cytometry) * Calculated creatinine clearance (using Cockcroft-Gault equation) \>= 30 mL/min * Absolute neutrophil count (ANC) \>= 1000/mm\^3 * Platelet count \>= 50,000/mm\^3 * Hemoglobin \>= 8.0 g/dL * Patients with measurable disease defined as at least one of the following: * For patients with extramedullary disease (EMD) measurable disease by computed tomography (CT) or magnetic resonance imaging (MRI) or the CT portion of the positron emission tomography (PET)/CT: must have at least one lesion that has a single diameter of \>= 2 cm; skin lesions can be used if the area is \>= 2 cm in at least one diameter and measured with a ruler * Plasma cell count \>= 0.5 X 10\^9/L or 5 percent of the peripheral blood white cells * Plasma cell count if determined by flow cytometry, \>= 200/150,000 events * Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0, 1 or 2 * Provide informed written consent * Negative pregnancy test done =\< 7 days prior to registration, for women of childbearing potential only * All study participants must be registered into the mandatory pomalidomide (POMALYST) Risk Evaluation and Mitigation Strategy (REMS) program, and be willing and able to comply with the requirements of the POMALYST REMS program * Willing to return to enrolling institution for follow-up (during the active monitoring phase of the study) * Willing to provide bone marrow and blood samples for correlative research purposes Exclusion Criteria: * Other malignancy requiring active therapy * EXCEPTIONS: Non-melanoma skin cancer, ductal breast carcinoma in situ (DCIS) or carcinoma-in-situ of the cervix * NOTE: If there is a history or prior malignancy, they must not be receiving other specific treatment for their cancer * Females of childbearing potential (FCBP)\* must have a negative serum pregnancy test with a sensitivity of at least 50 mIU/mL within 10-14 days prior to and again within 24 hours prior to prescribing pomalidomide for cycle 1 (prescriptions must be filled within 7 days as required by RevAssist \[lenalidomide REMS program\]), and must either commit to continued abstinence from heterosexual intercourse or begin TWO acceptable methods of birth control, one highly effective method and one additional effective method; AT THE SAME TIME, at least 28 days before she starts taking pomalidomide FCBP must also agree to ongoing pregnancy testing; men must agree to use a latex condom during sexual contact with a FCBP even if they have had a successful vasectomy; patient must follow pregnancy testing requirements as outlined in the POMALYST REMS program * A female of childbearing potential is a sexually mature woman who: 1) has not undergone a hysterectomy or bilateral oophorectomy; or 2) has not been naturally postmenopausal for at least 24 consecutive months (i.e., has had menses at any time in the preceding 24 consecutive months) * Nursing women * Men or women of childbearing potential who are unwilling to employ adequate contraception * Other co-morbidity which would interfere with patient's ability to participate in trial, e.g. uncontrolled infection, uncompensated heart or lung disease * Other concurrent chemotherapy, radiotherapy, or any ancillary therapy considered investigational * NOTE: Bisphosphonates are considered to be supportive care rather than therapy, and are thus allowed while on protocol treatment * Patient has \>= grade 3 peripheral neuropathy, or grade 2 with pain on clinical examination during the screening period * Major surgery =\< 14 days before study registration * Systemic treatment with strong CYP3A4 inducers (e.g. rifampin, rifapentine, rifabutin, carbamazepine, phenytoin, phenobarbital, gingko biloba, St. John's wort) within 7 days before registration * Evidence of current uncontrolled cardiovascular conditions, including cardiac arrhythmias, congestive heart failure, angina, or myocardial infarction within the past 6 months; Note: prior to study entry, any electrocardiogram (ECG) abnormality at screening must be documented by the investigator as not medically relevant * Corrected QT Interval (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 * 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 * Known gastrointestinal (GI) disease or GI procedure that could interfere with the oral absorption or tolerance of ixazomib including difficulty swallowing * Diarrhea \> grade 1, based on the National Cancer Institute Common Terminology Criteria for Adverse Events (NCI CTCAE) grading, in the absence of antidiarrheals