Anti-CD19/20/22 Chimeric Antigen Receptor T Cells (TriCAR19.20.22 T Cells) for the Treatment of Relapsed or Refractory Non-Hodgkin Lymphoma, Acute Lymphoblastic Leukemia, and Chronic Lymphocytic Leukemia

Inclusion Criteria: * COHORT A: Subjects must have relapsed or refractory non-Hodgkin lymphoma with lesions ≤ 5 cm, indolent lymphomas, or chronic lymphocytic leukemia without Richter's transformation * COHORT B: Subjects with lymphoid blast crisis from chronic myeloid leukemia, 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 * Subjects must have been treated with at least two lines of therapy; subjects with prior commercial or investigational CAR T therapy targeting CD19, and/or CD20, and/or CD22 are permitted. Disease must have either progressed after the last regimen or presented failure to achieve complete remission with the last regimen * Note: Cohort assignment at discretion of principal investigator (PI) depending on patient disease/ history * Subjects with relapsed/refractory CLL after at least 2 prior lines of appropriate therapy and must have previously received an approved Bruton's tyrosine kinase (BTK) inhibitor and venetoclax * In subjects who had a prior autologous stem cell transplant for refractory high-grade B-cell lymphoma who relapse within 12 months of autologous stem cell transplant are eligible * Subjects with relapsed/refractory acute B-lymphoblastic leukemia who received at least 2 prior lines of appropriate therapy. Subjects are also eligible if they have failed or are ineligible for allogeneic stem cell transplant * Subjects with relapsed/refractory lymphoid blast crisis from prior chronic myeloid leukemia (CML) who received at least 2 prior lines of therapy (tyrosine kinase inhibitors, multiagent chemotherapy) or have failed or are ineligible for allogeneic stem cell transplant * The patient's lymphoid malignancy must be positive for 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 * Subjects with prior commercial or investigational CAR T therapy targeting CD19, and/or CD20, and/or CD22 are permitted 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 * Subjects who received antibodies targeting CD19, or CD20, or CD22 are eligible * Age ≥ 18 years * Eastern Cooperative Oncology Group (ECOG) performance status ≤ 2 * Total bilirubin ≤ 1.5 times the institutional upper limit of normal * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\]) ≥ 3 x institutional upper limit of normal * Alanine aminotransferase (ALT) (serum glutamic pyruvic transaminase \[SGPT\]) ≤ 3 x institutional upper limit of normal * Creatinine clearance more than or equal to 50 ml/min calculated by the Cockcroft-Gault formula * 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; if white blood cell (WBC) is low and differential is not performed, CD3 count (helper/suppressor) should be ≥ 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 TriCAR19.20.22 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, as defined below: * 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 TriCAR19.20.22 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 TriCAR19.20.22 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 * Subjects with live vaccines given 28 days prior to lymphodepleting (LD) chemotherapy will be excluded * Active graft versus host disease * Active malignancy, other than non-melanoma skin cancer or carcinoma in situ (e.g. cervix, bladder, breast). Subjects 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 * 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 * Subjects 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 polymerase chain reaction (PCR) monitoring at the direction of an infectious disease specialist. Duration of prophylaxis to correspond with detection of TriCAR19.20.22 T cells/viral vector copies in serum or continued evidence of B-cell aplasia such as reduced intravenous immunoglobulin (IVIG) levels. No antiviral prophylaxis is indicated with hepatitis C positivity with negative PCR * Subjects 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