CD22 CAR T-cells to Extend Remission Following Commercial CD19 CAR T-cells in Children, Adolescents, and Adults With Relapsed/Refractory B-cell Acute Lymphoblastic Leukemia

* INCLUSION CRITERIA: * Participants must have documentation of pathologic confirmation of a diagnosis of relapsed/refractory B cell acute lymphoblastic leukemia (ALL). * History of CD19 and CD22 expression on malignant cells at diagnosis or relapse. * Age between \>= 3 years and \<= 65 years * Participants must have received an FDA-approved CD19 CAR T-cell construct for treatment of B cell ALL within the time period of \>= 2 months and \<= 7 months prior to apheresis or lymphodepleting (LD) (if apheresis is not done on this protocol). * Must be in an MRD-negative remission as demonstrated by flow cytometry at screening. * Must be ineligible for or unwilling to undergo allogeneic stem cell transplant (SCT). * Clinical performance status (PS): Karnofsky \>= 50% (participants \>= 16 years of age), or Lansky scale \>= 50% (participants \< 16 years of age). Participants who are unable to walk because of paralysis, but who are upright in a wheelchair may be considered eligible. * Must have no ongoing signs of CRS from prior CAR T cell infusion and/or ICANs at screening. * Participants must have adequate organ function as defined below: * Total bilirubin \<= 2 x institutional upper limit of normal (ULN) * Aspartate Aminotransferase (AST) \<= 10 x ULN * Alanine Aminotransferase (ALT) \<= 10 x ULN * creatinine \<= the maximum for age listed below OR measured creatinine clearance \>= 60 mL/min/1.73 m\^2 for participants with creatinine levels above the max * Age: \<=5, Maximum Serum, Creatinine \<= .8 mg/dL * Age: \>5 to \<=10, Maximum Serum, Creatinine \<= 1.0mg/dL * Age: \>10, Maximum Serum, Creatinine \<= 1.2mg/dL * A participant may have continued to expect CAR T cell-associated cytopenias of any grade. * Cardiac function: left ventricular ejection fraction\>= 45% or fractional shortening \>= 28%. * Pulmonary function: baseline oxygen saturation \>= 92% on room air; participants with respiratory symptoms (e.g., dyspnea, hypoxia \<92%) must have a diffusing capacity of the lungs for carbon monoxide (DLCO)/adjusted \> 45%. * Women of childbearing potential (WOCBP) must agree to use highly effective contraception (hormonal, intrauterine device (IUD), abstinence, surgical sterilization) at the study entry and up to 12 months after the last dose of combined chemotherapy. Note: WOCBP is defined as any individual who has experienced menarche and who has not undergone successful surgical sterilization or who is not postmenopausal. Men able to father a child must agree to use an effective method of contraception (barrier, surgical sterilization, abstinence) at the study entry and up to 7 months after the last dose of study drugs. We also will recommend men ask their partners to be on highly effective birth control (hormonal, IUD, surgical sterilization). Individuals able to father a child must not freeze or donate sperm within the same period. * Nursing participants must be willing to discontinue nursing from study treatment initiation through 6 months after the last dose of the study drug(s). * Participants must be enrolled on protocol 15-C-0028, Follow-Up Evaluation for Gene- Therapy Related Delayed Adverse Events after Participation in Pediatric Oncology Branch Clinical Trials. * Ability of participant or /Legally Authorized Representative (LAR) to understand and be willing to sign a written informed consent document. EXCLUSION CRITERIA: * Any central nervous system (CNS) involvement or signs of non-CNS extramedullary disease. * Any active graft versus host disease (GVHD) in participants who are post-HSCT. * Participants with disease recurrence requiring therapy post CD19 CAR. Note: Maintenance therapy post CD19 CAR (e.g., vincristine or tyrosine kinase inhibitor) for remission maintenance is allowed and will require a 1-week washout prior to apheresis or LD (if apheresis is not done on this protocol). * Any investigational agent within 1 week before apheresis or LD (if apheresis is not done on this protocol). * Pregnancy confirmed with beta-human chorionic gonadotropin (beta-HCG) serum or urine test performed at screening. * Human immunodeficiency virus (HIV) infection, as measured by seropositivity for (HIV) antibody. * Hepatitis B virus (HBV) infection, as measured by positivity for hepatitis B surface antigen (HBsAg). * Hepatitis C virus (HCV) infection, as measured by seropositivity for hepatitis C. * History of severe, immediate hypersensitivity reaction attributed to compounds of similar chemical or biological composition to any agent used in the study or in the manufacturing of cells. * Uncontrolled, symptomatic intercurrent illness evaluated by medical history, physical exam, and/or laboratory testing, or social situation that would limit compliance with study requirements or would pose an unacceptable risk to the participant.