Anti-CD19 CAR T-Cell Therapy for Blood Cancers

* INCLUSION CRITERIA: * Malignancy criteria * Histologically confirmed participants with either CLL or SLL or B-cell acute lymphoblastic leukemia or lymphoma (ALL) via immunohistochemical or flow cytometry methods will be eligible. Participants with evidence of Richter s transformation of CLL/SLL are also eligible. Participants with Richter s transformation must have current or prior evidence of CLL, confirmed by review of a current or prior histological sample by NIH pathologists or confirmed by flow cytometry performed at the NIH. * Demonstration of CD19 expression on CLL/SLL or ALL, as assessed by the NCI Laboratory of Pathology or NIH Department of Laboratory Medicine Hematopathology section. For participants with pathologically confirmed Richter s transformation, the transformed cells must also have CD19 expression. * CD19 expression must be uniform meaning no populations of clearly CD19-negative CLL/SLL, Richter s or ALL cells are observed. * CD20 must be detected on \>= 20% of malignant cells by flow cytometry or immunohistochemistry. Documentation of CD20 expression is not required for patients who have received CD20-directed therapy within 90 days prior to the date of enrollment. * The last dosage of systemic therapy (including corticosteroids) must be at least 14 days prior to the first dose of rituximab, with the exceptions of BTK inhibitors (BTKi) for CLL/SLL and tyrosine kinase inhibitors (TKI) for ALL. Participants who were receiving a BTKi for CLL/SLL or a TKI for ALL for at least 14 days prior to protocol enrollment can continue these agents during part of the time the participants are enrolled on this clinical trial. * For participants who have received antibodies targeting CD19, at least sixty days must elapse between therapy with antibodies targeting CD19 and CAR T-cell infusion. * Participants with CLL/SLL must have received at least two prior treatment regimens, at least one of which must have contained a Bruton s tyrosine kinase (BTK) inhibitor. Participants who took a BTK inhibitor but stopped due to intolerance are potentially eligible. Participants with relapsed or refractory CLL/SLL after alloHSCT are eligible. * Participants with refractory ALL that failed induction or participants with relapsed ALL after a standard induction regimen or after any later line of therapy are eligibleParticipants with relapsed or refractory ALL after alloHSCT are eligible. * All participants must have measurable malignancy as defined by at least one of the criteria below. * Presence of CLL,SLL, or ALL masses that are measurable (minimum 1.5 cm in largest diameter) by CT scan or PET/CT is required unless bone marrow or blood involvement with malignancy is detected. * For CLL/SLL or ALL with only bone marrow and/or blood involvement, no mass is necessary, but if a mass is not present, bone marrow and/or blood malignancy must be detectable by flow cytometry. Any level of CLL/SLL or ALL detectable by flow cytometry is sufficient. * Other inclusion criteria: * Age \>= 18 years. * Performance status (ECOG) 0-1. * Participants must have adequate organ and marrow function as defined below: * ANC \>= 1,000/mcL without the support of filgrastim or other growth factors in the 10 days prior to screening assessment * platelets \>= 50,000/mcL without transfusion support * hemoglobin \>= 8 g/dL * total bilirubin \<= 2.0 mg/dL * ALT or AST Serum ALT and AST less or equal to 3 times the upper limit of the institutional normal unless liver involvement by malignancy is demonstrated. If liver involvement with malignancy is detected, ALT and AST must be less than or equal to 5 times the upper limit of normal * Serum Creatinine Serum creatinine levels \< 1.5 X institutional ULN. Participants with serum creatinine \>= 1.5 X institutional ULN may participate if serum creatinine eGFR is \>=50 mL/min/1.73m\^2 by 2021 CKD-EPI equation. * ALT (SGPT)=alanine aminotransferase (serum glutamic pyruvic transaminase); * AST (SGOT)=aspartate aminotransferase (serum glutamic oxaloacetic transaminase); GFR=glomerular filtration rate; ULN=upper limit of normal. * (A)Creatinine clearance (CrCl) or eGFR should be calculated per institutional standard. * For CLL participants, B cells must make up less than 95% of blood lymphocytes on a lymphocyte phenotyping profile TBNK at the time of screening assessment. For ALL participants, the peripheral blood blast percentage on CBC differential must be 1% or less. * Room air oxygen saturation of 92% or greater * Participants of child-bearing or child-fathering potential must be willing to practice abstinence or highly effective contraception starting at the time of study entry, for the duration of study therapy, and for 12 months after receiving the protocol treatment. * Participants must agree not to donate eggs for 12 months after receiving the protocol treatment * Participants who are breastfeeding must be willing to cease breastfeeding from study treatment initiation through 12 months after the last dose of the study drugs. * Participants must have a negative blood PCR test for hepatitis B DNA. If hepatitis B DNA (PCR) testing is not available, participants must have a negative hepatitis B surface antigen and negative hepatitis B core antibody test. * Participants must have a negative blood PCR test for hepatitis C RNA. Only if Hepatitis C PCR testing is not available in a timely manner, participants must have a negative Hepatitis C antibody test. * Cardiac ejection fraction of greater than or equal to 50% by echocardiography and no evidence of hemodynamically significant pericardial effusion as determined by an echocardiogram within 30 days prior to treatment start. * All participants must have the ability to understand and willingness to sign a written informed consent. * All participants must be willing to undergo mandatory biopsies during the study. EXCLUSION CRITERIA: * Participants who are receiving any other investigational agents. * Participants who have had prior CAR T-cell therapy. * Participants who have had a live-attenuated or viral vector-based vaccine in the last 60 days prior to pre-leukapheresis rituximab. Participants who plan to receive a live attenuated or viral vector-based vaccine within the first 100 days after CAR T-cell infusion. * Participants that require urgent therapy due to tumor mass effects or spinal cord compression. * Current/active HIV infection, as measured by seropositivity for HIV antibody. * Participants with second malignancies in addition to their CLL or ALL are not eligible if the second malignancy has required treatment with surgery, radiation or chemotherapy, or other therapies within the past 2 years or is not in complete remission. Exceptions are that, in the last 2 years, participants may have had successful resection of non-metastatic basal cell or squamous cell carcinoma of the skin, and participants may have received hormonal therapy for fully resected breast cancer. * Positive beta Human chorionic gonadotropin (Beta-HCG) serum or urine pregnancy test in women of childbearing potential (WOCBP) performed at screening. * Active uncontrolled systemic infections (defined as infections causing fevers within 48 hours of the date of planned protocol rituximab or chemotherapy start and infections requiring intravenous antibiotics when intravenous antibiotics have been administered for less than 72 hours at the time of protocol rituximab or chemotherapy start). There must be objective evidence of infection, including, but not limited to, a positive blood, urine or sputum culture, positive nasal swab or blood test for viral infection, or the appearance of infiltrates on imaging of the lung. * Active coagulation disorders, major uncontrolled medical illnesses of the cardiovascular, respiratory, endocrine, renal, gastrointestinal, genitourinary or immune system, history of myocardial infarction, history of ventricular tachycardia or ventricular fibrillation, active cardiac arrhythmias with the exception of atrial fibrillation with baseline heart rates less than or equal to 90 beats per minute, (Use of medications to control heart rate is allowed.), active obstructive or restrictive pulmonary disease, or active autoimmune diseases such as rheumatoid arthritis. These include uncontrolled intercurrent illness manifesting as electrolyte derangements or as assessed by chemistries. * Significant neurologic disorders, including a history of a seizure disorder as an adult, that are not completely and permanently resolved and not requiring current treatment. * Any form of primary immunodeficiency (such as Severe Combined Immunodeficiency Disease) that has not been cured by a prior allogeneic stem cell transplant. * For participants that have not had prior allogeneic stem cell transplant: Systemic corticosteroid steroid therapy of any dose greater than 5 mg/day or more of prednisone or equivalent is not allowed within 14 days prior to the first dose of rituximab. Corticosteroid creams, ointments, and eye drops are allowed. * For participants that have had prior allogeneic stem cell transplant: Receiving any systemic immunosuppressive drugs including corticosteroids at doses of greater than 5 mg/day prednisone or equivalent within 28 days prior to Rituximab. Topical corticosteroid preparations applied to the skin such as solutions, creams, and ointments are allowed. Inhaled corticosteroids are allowed, and corticosteroid eye drops are allowed. * History of severe immediate hypersensitivity reaction to any of the agents used in this study, including hypersensitivity to aminoglycoside antibiotics, which may be used in the cell culture media. * Participants with CNS3 disease, neurologic signs of CNS disease, radiologically detected active CNS lymphoma or meningeal involvement. * Checkpoint inhibitor drugs such as pembrolizumab or nivolumab or other antibodies targeting PD-1 or PDL-1 within 180 days of pre-leukapheresis rituximab. This is because of possible effects checkpoint inhibitor therapy could have on the participant's T cells. * Known active alcohol or drug abuse. * History of allergy to study drug components. * Active tumor lysis syndrome as assessed by serum uric acid, LDH, calcium, and phosphorus. * Active rhabdomyolysis as assessed by elevated CK and acute change in renal function as reflected by increased creatinine and blood urea nitrogen (BUN). * Active diabetic ketoacidosis or hyperosmolar hyperglycemic state, as assessed by serum glucose. The urine will be tested for ketones if serum glucose is over 350 mg/dL at screening. * Participants who received a previous allogeneic HSCT must have no (grade 0) acute GVHD (Appendix H) and either no chronic GVHD or mild chronic GVHD. Participants with GVHD meeting the above criteria with local therapy (topical cutaneous steroids, inhaled steroids, and eye drops) will be eligible for enrollment.