CS1-CAR T Therapy Following Chemotherapy in Treating Patients With Relapsed or Refractory CS1 Positive Multiple Myeloma

Inclusion Criteria: * Documented informed consent of the participant and/or legally authorized representative. * Assent, when appropriate, will be obtained per institutional guidelines. * Karnofsky Performance Status (KPS) of \>= 70%. * Life expectancy \>= 16 weeks. * Participant must have a confirmed diagnosis of active MM as defined by the International Myeloma Working Group (IMWG) criteria. * Participant must have a confirmed CS1+ MM as evaluated by City of Hope (COH) Pathology Core. * Participant must have measurable disease defined as meeting at least one of the criteria below: * Serum M-protein \>= 0.5 g/dL. * Urine M-protein \>= 200 mg/24 hour. * Involved serum free light chain (sFLC) level \>= 10 mg/dL with abnormal kappa/lambda ratio. * Measurable biopsy-proven plasmacytomas (\>= 1 lesion that has a single diameter \>= 2 cm) * Bone marrow plasma cells \>= 30%. * Participant must have relapsed or refractory disease after all 3 prior treatment regimens with the following requirements: * Participant must have received prior treatment with an immunomodulatory agent. * Participant must have received prior treatment with a proteasome inhibitor. * Participant must have received prior treatment with an anti-CD38 antibody. * Participants must be refractory to last line of therapy prior to study enrollment (refractory myeloma is defined as disease that is nonresponsive, progression on treatment, or shows progression within 60 days after the last prior line of therapy). * Participants who were not candidates to receive one or more of the above treatments are eligible; however, the reason must be clearly documented in the case report form. * Note: induction chemotherapy, autologous stem-cell transplantation (ASCT), and maintenance therapy should be considered as 1 "regimen." * Additionally, if a participation underwent autologous transplant he/she be \>= 90 days from transplant at the time of enrollment. * Total serum bilirubin =\< 2.0 mg/dL. * Participants with Gilbert syndrome may be included if their total bilirubin is =\< 3.0. * Aspartate aminotransferase (AST) \< 2.5 x upper limit of normal (ULN). * Alanine aminotransferase (ALT) \< 2.5 x ULN. * Serum creatinine =\< 2.5 x ULN or estimated creatinine clearance of \>= 40 mL/min per the Cockcroft-Gault formula, and the participant is not on hemodialysis. * Absolute neutrophil count \>= 1000/uL. Transfusions and growth factors must not be used to meet these requirements at initial screening. * Hemoglobin (Hb) \>= 8 g/dl. Transfusions and growth factors must not be used to meet these requirements at initial screening. * Platelet count \>= 50,000/uL (\>= 30,000/uL if bone marrow plasma cells are \>= 50% of cellularity). Transfusions and growth factors must not be used to meet these requirements at initial screening. * Left ventricular ejection fraction \>= 45% within 8 weeks before enrollment. * Oxygen (O2) saturation \>= 92%. * 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 3 months 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 allogeneic stem cell transplantation. * Autologous transplantation =\< 90 days of enrollment. * Growth factors within 14 days of enrollment. * Platelet transfusions within 7 days of enrollment. * Epstein-Barr virus (EBV) positivity by polymerase chain reaction (PCR) at the time of enrollment * Participants receiving any other investigational agents, or concurrent biological, chemotherapy, or radiation therapy. * Participants with known additional malignancy that is progressing or required active treatment within the last 2 years. Exceptions include basal cell carcinoma of the skin, squamous cell carcinoma of the skin, or in situ cervical cancer that has undergone potentially curative therapy. * Participants with toxicities from prior therapies, with the exception of peripheral neuropathy attributable to bortezomib, that have not recovered to grade =\< 2 according to the Common Terminology Criteria for Adverse Events (CTCAE) 5.0 criteria or to the subject's prior baseline. * Participants with known active hepatitis B or C infection; research participants who are human immunodeficiency virus (HIV) positive based on testing performed within 4 weeks of enrollment; research participants with any signs or symptoms of active infection, positive blood cultures or radiological evidence of infections. * Participants with active auto-immune disease, including connective tissue disease, sarcoidosis, multiple sclerosis, inflammatory bowel disease or have a history of severe (as judged by the principal investigator) autoimmune disease that will require prolonged immunosuppressive therapy. * Have New York Heart Association (NYHA) Class III or IV heart failure, unstable angina, or a history of recent (within 6 months) myocardial infarction. * Participants with a history or presence of clinically relevant central nervous system (CNS) pathology such as uncontrolled seizure disorder, stroke, severe brain injuries, dementia, cerebellar disease or psychosis. * Participants with known active central nervous system (CNS) involvement by malignancy. Subjects with prior CNS disease that has been effectively treated will be eligible if treatment was completed at least 3 months prior to enrollment with no evidence of symptomatic disease and stable abnormalities on repeat imaging. * Participants with plasma cell leukemia (PCL) or symptomatic amyloidosis. However, participants with a prior history of PCL are not excluded. * Participants with any known contraindications to leukapheresis, cyclophosphamide, fludarabine, cetuximab or tocilizumab. * Dependence on corticosteroids. * Defined as doses of corticosteroids of greater than or equal to 10 mg/day of prednisone or equivalent doses of other corticosteroids. * Note: Topical and inhaled corticosteroids in standard doses and physiologic replacement for subjects with adrenal insufficiency are allowed. * Participants with inadequate venous access for leukapheresis, and who are either unable to or unwilling to have a supportive line (temporary or other) placed for the procedure. * Females only: Pregnant or breastfeeding. * Any other condition that would, in the Investigator's judgment, contraindicate the subject's participation in the clinical study due to safety concerns with clinical study procedures. * Prospective subjects who, in the opinion of the investigator, may not be able to comply with all study procedures (including compliance issues related to feasibility/logistics).