Immunotherapy With BCMA CAR-T Cells in Treating Patients With BCMA Positive Relapsed or Refractory Multiple Myeloma

Inclusion Criteria:

* Have the capacity to give informed consent
* Eastern Cooperative Oncology Group (ECOG) performance status score =\< 2.
* 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 kappa/lambda ratio
  * Measurable biopsy-proven plasmacytomas (\>= 1 lesion that has a single diameter \>= 2 cm)
  * Bone marrow plasma cells \>= 30%
* Have a diagnosis of BCMA+ 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 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 principal investigator 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; \> 4 cycles of therapy are not required for patients with a diagnosis of plasma cell leukemia

      * Patients receiving retreatment do not need to meet the \> 10% CD138+ malignant plasma cells (immunohistochemistry staining method \[IHC\]) on BM core biopsy
* Male and female patients of reproductive potential must be willing to use an effect contraceptive method before, during, and for at least 4 months after the CAR T cell infusion

Exclusion Criteria:

* History of another primary malignancy that requires intervention beyond surveillance or that has not been in remission for at least 1 year (the following are exempt from the 1 year limit: non-melanoma skin cancer, curatively treated localized prostate cancer, and cervical carcinoma in situ on biopsy or a squamous intraepithelial lesion on papanicolaou \[PAP\] smear)
* Active hepatitis B, hepatitis C at the time of screening
* Patients who are (human immunodeficiency virus \[HIV\]) seropositive
* Subjects with uncontrolled active infection
* \> 1 hospital admission (lasting 5 days or more) for documented infection in prior 6 months
* Presence of acute or chronic graft-versus-host disease (GVHD) requiring active treatment 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 as determined by the principal investigator (PI) or designee
* History of clinically relevant or active central nervous system (CNS) pathology such as epilepsy, seizure, paresis, aphasia, stroke, severe brain injuries, dementia, Parkinson's disease, cerebellar disease, organic brain syndrome, or 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 14 days of enrollment) and have no evidence of new sites of CNS activity
* Pregnant or breastfeeding females
* Allogeneic HSCT or donor lymphocyte infusion within 90 days of leukapheresis.
* 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
  * Cytotoxic chemotherapeutic agents within 1 week of leukapheresis; oral chemotherapeutic agents are allowed if at least 3 half-lives have elapsed prior to leukapheresis
  * Lymphotoxic chemotherapeutic agents within 2 weeks of leukapheresis
  * Daratumumab (or other anti-CD38 therapy) within 30 days 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
* Absolute neutrophil count (ANC) \< 1000/mm\^3, or per PI discretion if cytopenia thought to be related to underlying myeloma.
* Hemoglobin (Hgb) \< 8 mg/dl, or per PI discretion if cytopenia thought to be related to underlying myeloma.
* Platelet count \< 50,000/mm\^3, or per PI discretion if cytopenia thought to be related to underlying myeloma.
* Active autoimmune disease requiring immunosuppressive therapy
* Major organ dysfunction defined as:

  * Creatinine clearance \< 20 ml/min
  * Significant hepatic dysfunction (serum glutamic-oxaloacetic transaminase \[SGOT\] \> 5 x 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
* Uncontrolled medical, psychological, familial, sociological, or geographical conditions that do not permit compliance with the protocol, as judged by the PI; or unwillingness or inability to follow the procedures required in the protocol