Leflunomide, Pomalidomide, and Dexamethasone for the Treatment of Relapsed or Refractory Multiple Myeloma

Inclusion Criteria: * Documented informed consent of the participant and/or legally authorized representative * Assent, when appropriate, will be obtained per institutional guidelines * Agreement to allow the use of archival tissue from diagnostic tumor biopsies * If unavailable, exceptions may be granted with study principal investigator (PI) approval * Eastern Cooperative Oncology Group (ECOG) =\< 2 * Life expectancy \> 3 months * Diagnosis of multiple myeloma with measurable disease as defined by: * M-protein quantities \>= 0.5 g/dL by serum protein electrophoresis (sPEP) or * \>= 200 mg/24 hour urine collection by urine protein electrophoresis (uPEP) or * Serum free light chain (FLC) \> 10.0 mg/dL involved light chain and an abnormal kappa/lambda ration in subjects without detectable serum or urine M-protein or * For subjects with immunoglobulin class A (IgA) myeloma whose disease can only be reliably measured by quantitative immunoglobulin measurement, a serum IgA level \>= 0.50 g/dL * Relapsed or refractory to at least 1 prior line of therapy, including both a proteasome inhibitor and an immunomodulatory drug, and for whom transplant is not recommended. Participants may opt for a delayed transplant at a later time, if appropriate * Fully recovered from the acute toxic effects (except alopecia) to =\< grade 2 to prior anti-cancer therapy * Absolute neutrophil count (ANC) \>= 1.0 x 10\^9/L (performed within 30 days prior to day 1 of protocol therapy unless otherwise stated) * NOTE: Screening ANC should be independent of granulocyte- and granulocyte/macrophage colony stimulating factor (G-CSF and GM-CSF) support for at least 1 week and of pegylated G-CSF for at least 2 weeks * Platelets \>= 75.0 x 10\^9/L (performed within 30 days prior to day 1 of protocol therapy unless otherwise stated) * NOTE: Screening platelet count should be independent of platelet transfusions for at least 2 weeks * Hemoglobin \>= 8.0 g/dL (performed within 30 days prior to day 1 of protocol therapy unless otherwise stated) * NOTE: Transfusion support is allowed * Total bilirubin =\< 2 X upper limit of normal (ULN) (unless has Gilbert's disease) (performed within 30 days prior to day 1 of protocol therapy unless otherwise stated) * Aspartate aminotransferase (AST) =\< 3.5 x ULN (performed within 30 days prior to day 1 of protocol therapy unless otherwise stated) * Alanine aminotransferase (ALT) =\< 3.5 x ULN (performed within 30 days prior to day 1 of protocol therapy unless otherwise stated) * Alkaline phosphatase \< 5 x ULN (performed within 30 days prior to day 1 of protocol therapy unless otherwise stated) * Creatinine clearance of \>= 30 mL/min per 24 hour urine test (performed within 30 days prior to day 1 of protocol therapy unless otherwise stated) * Women of childbearing potential (WOCBP): negative urine or serum pregnancy test * If the urine test is positive or cannot be confirmed as negative, a serum pregnancy test will be required * Agreement by females and males of childbearing potential to use an effective method of birth control or abstain from heterosexual activity for the course of the study through at least 4 weeks after the last dose of protocol therapy * Childbearing potential defined as not being surgically sterilized (men and women) or have not been free from menses for \> 1 year (women only) Exclusion Criteria: * Prior treatment with leflunomide * Patients who are pomalidomide refractory, defined as subjects who progress on or within 60 days of pomalidomide when given as a single agent or in combinatorial therapies. Prior exposure to pomalidomide without refractoriness is allowed * Current or planned use of other anti-myeloma therapies besides leflunomide, pomalidomide, and dexamethasone * Current or planned growth factor or transfusion support until after initiation of treatment * Prior allogeneic transplant * History of allergic reactions attributed to compounds of similar chemical or biologic composition to study agents * Positive for tuberculosis or latent tuberculosis (TB) * Positive for hepatitis A, B, or C * Known human immunodeficiency virus (HIV) infection * Prior diagnosis of rheumatoid arthritis * Acute active infection requiring systemic therapy within 2 weeks prior to enrollment * Subject has history of anaphylaxis to thalidomide, lenalidomide, pomalidomide, cholestyramine or dexamethasone * Non-hematologic malignancies within the past 3 years, with the exceptions of * Adequately treated basal cell or squamous cell skin cancer, * Carcinoma in situ of the cervix, * Prostate cancer \< Gleason grade 6 with stable prostate specific antigen (PSA), or * Successfully treated in situ carcinoma of the breast * Females only: Pregnant or breastfeeding * Any other condition that would, in the Investigator's judgment, contraindicate the patient's participation in the clinical study due to safety concerns with clinical study procedures * Prospective participants who, in the opinion of the investigator, may not be able to comply with all study procedures (including compliance issues related to feasibility/logistics)