CART-BCMA/CS1 in Treating Patients With Relapsed or Refractory Multiple Myeloma

Inclusion Criteria: * Diagnosis of multiple myeloma relapsed or refractory after at least three prior lines of therapy, including: * A proteasome inhibitor and immunomodulatory agent either alone or in combination * Anti-CD38 (Cluster of Differentiation 38) directed therapy * Patients previously treated with anti-BCMA directed therapy are allowed onto the study. Patients must not have a history of grade \> 3 CRS or grade \>= 3 immune effector cell associated neurotoxicity (ICANS) with prior anti-BCMA therapy * Patients must have measurable MM as defined by at least one of the criteria below: * Serum M-protein \>= 0.5 g/dl (5 g/L) * Urine M-protein \>= 200 mg/24 h * Serum free light chain (FLC) assay: involved FLC level \>= 10 mg/dl (100 mg/l) provided serum FLC ratio is abnormal * A biopsy-proven plasmacytoma * Age \>= 18 years old and =\< 74 years old * Eastern Cooperative Oncology Group (ECOG) performance status 0 or 1 * Absolute neutrophil count (ANC) \>= 1 x 10\^9 cells/L. Granulocyte colony stimulating factor is permitted (within 30 days prior to enrollment using standard phase I criteria for organ function) * Platelets \>= 50 x 10\^9/L. Without transfusion, growth factors may be used to achieve eligibility criteria (within 30 days prior to enrollment using standard phase I criteria for organ function) * Hemoglobin \>= 8 g/dL (with or without transfusion) (within 30 days prior to enrollment using standard phase I criteria for organ function) * Aspartate and alanine aminotransferases (AST, ALT) =\< 2.5 x upper limit of normal (ULN) (within 30 days prior to enrollment using standard phase I criteria for organ function) * Total bilirubin =\< 2 x ULN (except patients with documented Gilbert's syndrome) (Within 30 days prior to enrollment using standard phase I criteria for organ function) * Creatinine clearance \>= 30 mL/min (within 30 days prior to enrollment using standard phase I criteria for organ function) * Must be willing and able to accept at least one leukapheresis procedure * Must be willing and able to provide written informed consent Exclusion Criteria: * Inability to purify \>= 5 x 10\^8 CD62L-enriched cells from leukapheresis product * Previously known hypersensitivity to any of the agents used in this study; known sensitivity to cyclophosphamide or fludarabine * Received systemic treatment for multiple myeloma, including immunotherapy, within 14 days prior to initiation of lymphodepletion chemotherapy administration within this protocol. Consistent with current trials, patients may otherwise be given bridging therapy at the discretion of the lead study investigator * Prior allogeneic hematopoietic stem cell transplantation * Autologous hematopoietic stem cell transplantation within 12 weeks prior to enrollment. Patients who have received an autologous transplant over 12 weeks from enrollment will not be excluded and may participate in the trial * Potential requirement for systemic corticosteroids or concurrent immunosuppressive drugs based on prior history or having received systemic steroids within the last 2 weeks prior to enrollment (inhaled or topical steroids at standard doses are allowed) * Human immunodeficiency virus (HIV) seropositivity or other congenital or acquired immune deficiency state, which would increase the risk of opportunistic infections and other complications during chemotherapy-induced lymphodepletion. If there is a positive result in the infectious disease testing that was not previously known, the patient will be referred to their primary physician and/or infectious disease specialist * Hepatitis B or C seropositivity with evidence of ongoing liver damage, which would increase the likelihood of hepatic toxicities from the lymphodepletion chemotherapy regimen and supportive treatments. Patients with hepatitis C seropositive disease but undetectable hepatic C virus (HCV) ribonucleic acid (RNA) viral load will be allowed in the trial. Patients with B seropositivity on antiviral therapy will be allowed in the trial * Dementia or significantly altered mental status that would prohibit the understanding or rendering of informed consent and compliance with the requirements of this protocol * Known clinically active central nervous system involvement (CNS). Prior evidence of CNS involvement successfully treated will not be an exclusion for participation as long as they are deemed under control at the time of study enrollment and there are no neurological signs of potential CNS involvement. A brain magnetic resonance imaging (MRI) scan taken within 8 weeks of lymphodepletion may be used, otherwise a brain MRI must be performed to confirm absence of CNS involvement * Oxygen saturation of =\< 92% on room air * A left ventricular ejection fraction =\< 45% * Pregnancy or breast-feeding. Female patients must be surgically sterile or be postmenopausal for two years or must agree to use effective contraception during the period of treatment and for 6 months afterwards. All female patients with reproductive potential must have a negative pregnancy test (serum/urine) at screening and again within 14 days from starting the lymphodepletion chemotherapy. The definition of effective contraception will be based on the judgment of the study investigators. Patients who are breastfeeding are not allowed on this study * History of other malignancy in the past 3 years with the following exceptions: * Malignancy treated with curative intent and no known active disease * Adequately treated non-melanoma skin cancer without evidence of disease * Adequately treated cervical carcinoma in situ without evidence of disease * Adequately treated breast ductile carcinoma without evidence of disease * Prostate cancer with a Gleason score less than 6 with undetectable prostate specific antigen over 12 months * Adequately treated urothelial non-invasive carcinoma or carcinoma in situ * Similar neoplastic conditions with an expectation of greater than 95% disease free survival