Anti-BCMA or/and Anti-CD19 CART Cells Treatment of Relapsed Multiple Myeloma

Inclusion Criteria: * Have the capacity to give informed consent * Have measurable disease by International Myeloma Working Group (IMWG) criteria based on one or more of the following findings: * Serum M-protein \>= 1 g/dL * Urine M-protein \>= 200 mg/24 hour * Involved serum free light chain (sFLC) level \>= 10 mg/dL with abnormal κ/λ ratio * Measurable biopsy-proven plasmacytomas (\>= 1 lesion that has a single diameter \>= 2 cm) * Bone marrow plasma cells \>= 30% * Have a diagnosis of BCMA+ multiple myeloma (MM) (\>= 5% BCMA+ by flow cytometry on CD138 co-expressing plasma cells obtained within 45 days of study enrollment); the MM diagnosis must be confirmed by internal pathology review of a fresh biopsy specimen at the Fred Hutchinson Cancer Research Center (FHCRC)/Seattle Cancer Care Alliance (SCCA) * Have relapsed or treatment refractory disease with \>= 10% CD138+ malignant plasma cells (IHC) on bone marrow (BM) core biopsy, either: * Following autologous stem cell transplant (ASCT) * Or, if a patient has not yet undergone ASCT, the individual must: * Be transplant ineligible, due to age, comorbidity, patient choice, insurance reasons, concerns of rapidly progressive disease, and/or discretion of attending physician, and, * Demonstrate disease that persists after \> 4 cycles of induction therapy and that is double refractory (persistence/progression) after therapy with both a proteasome inhibitor and immunomodulatory drug (IMiD) administered either in tandem, or in sequence. Exclusion Criteria: * Active hepatitis B, hepatitis C at the time of screening * Patients who are (human immunodeficiency virus \[HIV\]) seropositive * Subjects with uncontrolled systemic fungal, bacterial, viral or other infection despite appropriate antibiotics or other treatment at the time of leukapheresis * \> 1 hospital admission for infection in prior 3 months * Presence of acute or chronic graft-versus-host disease (GVHD) unless limited to skin involvement and managed with topical steroid therapy alone * History of any one of the following cardiovascular conditions within the past 6 months: class III or IV heart failure as defined by the New York Heart Association (NYHA), cardiac angioplasty or stenting, myocardial infarction, unstable angina, or other clinically significant cardiac disease * History or presence of clinically relevant central nervous system (CNS) pathology such as epilepsy, seizure, paresis, aphasia, stroke, severe brain injuries, dementia, Parkinson's disease, cerebellar disease, organic brain syndrome, psychosis, active central nervous system MM involvement and/or carcinomatous meningitis; subjects with previously treated central nervous systems involvement may participate, provided they are free of disease in the CNS (documented by flow cytometry performed on the cerebrospinal fluid (CSF) within one week of enrollment) and have no evidence of new sites of CNS activity * Pregnant or nursing women; NOTE: Women of reproductive potential must have a negative serum pregnancy test performed within 48 hours of starting conditioning chemotherapy * Use of any of the following: * Therapeutic doses of corticosteroids (defined as \> 20 mg/day prednisone or equivalent) within 7 days prior to leukapheresis; physiologic replacement, topical, and inhaled steroids are permitted * Allogeneic hematopoietic stem cell transplant (allo-HSCT) within 90 days of leukapheresis * Cytotoxic chemotherapeutic agents within 1 week of leukapheresis; oral chemotherapeutic agents are allowed if at least 3 half-lives have elapsed prior to leukapheresis * Low dose chemotherapy (e.g., bortezomib, lenalidomide, cyclophosphamide =\< 300 mg/m\^2) given after leukapheresis to maintain disease control must be stopped \>= 7 days prior to initiation of lymphodepleting chemotherapy * Lymphotoxic chemotherapeutic agents within 2 weeks of leukapheresis * Experimental agents within 4 weeks of leukapheresis unless progression is documented on therapy and at least 3 half-lives have elapsed prior to leukapheresis * Uncontrolled medical, psychological, familial, sociological, or geographical conditions that do not permit compliance with the protocol, as judged by the investigator; or unwillingness or inability to follow the procedures required in the protocol * Absolute neutrophil count (ANC) \< 1.0×10E9/L, Hemoglobin (Hgb) \< 80 g/L, Platelet count \< 50×10E9/L * Active autoimmune disease requiring immunosuppressive therapy * Major organ dysfunction defined as: * Creatinine clearance \< 20 ml/min * Significant hepatic dysfunction (serum glutamic-oxaloacetic transaminase \[SGOT\] or serum glutamic-oxaloacetic transaminase (SGOT) or serum alanine aminotransferase (SALT) \> 5×upper limit of normal; bilirubin \> 3.0 mg/dL) * Forced expiratory volume in 1 second (FEV1) of \< 50% predicted or diffusion capacity of the lung for carbon monoxide (DLCO) (corrected) \< 40% (patients with clinically significant pulmonary dysfunction, as determined by medical history and physical exam should undergo pulmonary function testing) * Anticipated survival of \< 3 months * Contraindication to cyclophosphamide or fludarabine chemotherapy * Patients with known AL subtype amyloidosis