Genetically Engineered Cells (Anti-CD19/CD20/CD22 CAR T-cells) for the Treatment of Relapsed or Refractory Lymphoid Malignancies

Inclusion Criteria: * Adult subjects with relapsed or refractory non-Hodgkin lymphoma with lesions =\< 5 cm, indolent lymphomas, chronic lymphocytic leukemia without Richter's transformation, or B-prolymphocytic leukemia (Cohort A) * OR adult subjects with lymphoid blast crisis, acute lymphoblastic leukemia, chronic lymphocytic leukemia with Richter's transformation, non-Hodgkin lymphoma with lesions \> 5 cm and/or lymphoblastic lymphoma, or non-Hodgkin lymphoma with circulating lymphoma cells, B-Prolymphocytic leukemia with lesions \> 5 cm (not including splenomegaly (Cohort B). * OR Pediatric subjects with Acute Lymphoblastic Leukemia or Non-Hodgkin Lymphoma * Subjects must have been treated with at least two lines of therapy. Disease must have either progressed after the last regimen or presented failure to achieve complete remission with the last regimen. B-PLL is defined as having greater than 55% prolymphocytes in the peripheral blood * Subjects with relapsed/refractory CLL after at least 2 prior lines of appropriate therapy and must have previously received an approved BTK inhibitor and venetoclax * Subjects with refractory high-grade B-cell lymphoma who relapse within 12 months of autologous stem cell transplant * Subjects with relapsed/refractory B-prolymphocytic leukemia who received at least 1- 2 prior lines of appropriate therapy and who have failed or are ineligible for allogeneic stem cell transplant * Subjects with relapsed/refractory acute B-lymphoblastic leukemia who received at least 2 prior lines of appropriate therapy or who have failed or are ineligible for allogeneic stem cell transplant. * The patient's lymphoid malignancy must be positive for at least one target antigen (CD19 and/or CD20 and/or CD22), either by immunohistochemistry or flow cytometry analysis on the last biopsy available or peripheral blood for circulating disease. * Patients who received blinatumomab or inotuzumab are eligible. * Patients who received prior CAR T-cells are eligible, (commercial CD 19 CAR-T cells or dual CAR-T cells), if it has been at least 30 days since previous CAR T cell therapy and \<5% of circulating levels of CD3+ cells express the prior CAR by flow cytometry. * Age \>= 2 years * Eastern Cooperative Oncology Group (ECOG) performance status =\< 2. For patients \< 16 years, Performance score Lansky \>= 50 * Total bilirubin =\< 1.5 times the institutional upper limit of normal for age * Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase \[SGOT\]) =\< 3 X institutional upper limit of normal for age * Alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 3 X institutional upper limit of normal for age * Creatinine clearance more than or equal to 50 ml/min calculated by the Cockcroft - Gault formula, or by Schwartz formula for patients \< 18 years * Subjects must have adequate pulmonary function as defined as pulse oximetry \>= 92% on room air * Subjects must have adequate cardiac function as defined as left ventricular ejection fraction \>= 40% in the most recent echocardiogram * Absolute lymphocyte count \> 100/uL * Subjects (or legal guardians) must have the ability to understand and the willingness to sign a written informed consent document * For women of childbearing potential: agreement to remain abstinent (refrain from heterosexual intercourse) or use a contraceptive method with a failure rate of \< 1% per year during the treatment period and for at least 6 months after the Anti-CD19/20/22 CAR-T cell infusion * A woman is considered to be of childbearing potential if she is postmenarcheal, has not reached a postmenopausal state (\< 12 continuous months of amenorrhea with no identified cause other than menopause), and has not undergone surgical sterilization (removal of ovaries and/or uterus). Examples of contraceptive methods with a failure rate of \< 1% per year include bilateral tubal ligation, male sterilization, hormonal contraceptive s that inhibit ovulation, hormone-releasing intrauterine devices, and copper intrauterine devices * The reliability of sexual abstinence should be evaluated in relation to the duration of the clinical trial and the preferred and usual lifestyle of the patient. Periodic abstinence (e.g., calendar, ovulation, symptothermal, or postovulation methods) and withdrawal are not acceptable methods of contraception * For men: agreement to remain abstinent (refrain from heterosexual intercourse) or use contraceptive measures, and agreement to refrain from donating sperm * With female partners of childbearing potential, men must remain abstinent or use a condom plus an additional contraceptive method that together result in a failure rate of \< 1% per year during the treatment period and for at least 6 months after the Anti-CD19/20/22 CAR-T cell infusion. Men must refrain from donating sperm during this same period * With pregnant female partners, men must remain abstinent or use a condom during the treatment period and for at least 6 months after the human anti-CD19 CAR-T cell infusion to avoid potential embryonal or fetal exposure. The reliability of sexual abstinence should be evaluated in relation to the duration of the clinical trial and the preferred and usual lifestyle of the patient. Periodic abstinence (e.g., calendar, ovulation, symptothermal, or postovulation methods) and withdrawal are not acceptable methods of contraception. Exclusion Criteria: * Autologous transplant within 6 weeks of planned CAR-T cell infusion * Allogeneic stem cell transplant or donor lymphocyte infusion within 2 months of planned CAR-T cell infusion and patients must be off immunosuppressive agents. Patients with live vaccines given 28 days prior to lymphodepletion (LD) chemotherapy will be excluded * Active graft versus host disease * Active central nervous system or meningeal involvement by lymphoma or leukemia. Subjects with untreated brain metastases/central nervous system (CNS) disease will be excluded from this clinical trial because of their poor prognosis and because they often develop progressive neurologic dysfunction that would confound the evaluation of neurologic and other adverse events. Patients with a history of CNS or meningeal involvement must be in a documented remission by CSF evaluation and contrast-enhanced magnetic resonance imaging (MRI) imaging for at least 90 days prior to registration * Active malignancy, other than non-melanoma skin cancer or carcinoma in situ (e.g.cervix, bladder, breast). Patients with a prior or concurrent malignancy whose natural history or treatment does not have the potential to interfere with the safety or efficacy assessment of the investigational regimen are eligible for this trial (e.g. Low Gleason score prostate Cancer) * A minimum of 28 days must have elapsed between prior treatment with investigational agent(s) and the day of lymphocyte collection * Human immunodeficiency virus (HIV)-seropositive patients are allowable, however must be on effective anti-retroviral therapy with undetectable viral load within 6 months of enrollment to be eligible for this trial * Subjects with uncontrolled intercurrent illness including, but not limited to ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, pulmonary abnormalities or psychiatric illness/social situations that would limit compliance with study requirements * Pregnant or breastfeeding women are excluded from this study because CAR-T cell therapy may be associated with the potential for teratogenic or abortifacient effects. Women of childbearing potential must have a negative serum pregnancy test. Because there is an unknown, but potential risk for adverse events in nursing infants secondary to treatment of the mother with CAR-T cells, breastfeeding should be discontinued. These potential risks may also apply to other agents used in this study * Evidence of myelodysplasia or cytogenetic abnormality indicative of myelodysplasia on any bone marrow biopsy prior to initiation of therapy * Patients with a positive hepatitis B core antibody or surface antigen are at high risk for hepatitis B virus (HBV) reaction and will require entecavir/tenofivir prophylaxis or serial hepatitis B (Hep B) PCR monitoring at the direction of an infectious disease specialist. Duration of prophylaxis to correspond with detection of Anti-CD19/20/22 CAR T cells/viral vector copies in serum or continued evidence of B-cell aplasia such as reduced intravenous immunoglobulin therapy (IVIG) levels. No antiviral prophylaxis is indicated with hepatitis C positivity with negative PCR * Patients with history of clinically relevant CNS pathology such as epilepsy, seizure disorders, paresis, aphasia, uncontrolled cerebrovascular disease, severe brain injuries, dementia and Parkinson's disease * History of autoimmune disease (i.e. rheumatoid arthritis, systemic lupus erythematosus) with requirement of immunosuppressive medication within 6 months * Live vaccines given in 28 days prior to lymphodepleting chemotherapy