• 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.