A Multi-center Single Arm Phase II Study to Evaluate the Safety and Efficacy of Genetically Engineered Autologous Cells Expressing Anti-CD20 and Anti-CD19 Specific Chimeric Antigen Receptor in Subjects With Relapsed and/or Refractory Diffuse Large B Cell Lymphoma

Inclusion Criteria: * Histologically confirmed B-cell non-Hodgkin's lymphoma: * DLBCL DLBCL or associated subtype, defined by WHO 2016 classification: * DLBCL not otherwise specified (NOS) * High-grade B cell lymphoma with MYC and BCL2 and/or BCL6 rearrangements * High-grade B cell lymphoma (NOS) * Primary mediastinal (thymic) large B cell lymphoma * Transformed lymphoma (e.g., transformed follicular, or marginal zone lymphoma, follicular lymphoma (FL Grade 3) * CNS Cohort only: B-cell primary or secondary central nervous system lymphoma (PCNSL or SCNSL) * Mantle Cell Lymphoma (MCL) Cohort: Histologically confirmed MCL determined by overexpression of cyclin D1 or presence of t(11;14) (q13; q32) translocation * Richter's Transformation (RT) Cohort: Histologically confirmed Richter's transformation (RT) to a diffuse large B-cell lymphoma (DLBCL) subtype from underlying CLL (clonally related) * Relapsed or refractory disease is defined for DLBCL (and associated subtypes) population as failure of 2 or more lines of chemotherapy including rituximab or equivalent and anthracycline and either having failed autologous stem cell transplant (ASCT), or ineligible, not intended for or not consenting to ASCT * Chemotherapy-refractory disease is defined as persistent disease after last line of therapy or relapsed or persistent disease after prior ASCT for lymphoma * Disease relapse in subjects without prior ASCT is defined as relapse of disease after the last dose of most recent therapy regimen * CNS Cohort: Subjects with relapsed/refractory PCNSL that have failed (or unable to tolerate) at least first-line therapy. * No contraindications for MRI evaluation * CNS Cohort: Subjects with SCNSL must have relapsed or refractory disease after having received at least one prior line of systemic therapy * Prior lines of systemic therapy should include an anti-CD20 monoclonal antibody and anthracycline containing chemotherapy regimen and/or with or without an autologous stem cell transplant * No contraindications for MRI evaluation * MCL Cohort: Subjects with relapsed/refractory disease after at least one prior systemic treatment, that must include: * Cytotoxic rituximab-based chemotherapy regimen (eg, rituximab bendamustine, R-CHOP, R-DHAP, R-ARA-C) AND * BTK inhibitor * RT Cohort: Subject must have relapsed/refractory disease after at least one prior systemic treatment following Richter's Transformation * Age ≥18 years * Eastern Cooperative Oncology Group (ECOG) performance status that is either 0 or 1 at screening. ECOG performance status of 2 at screen is allowed if the decrease in performance status is due to lymphoma * Measurable disease according to Lugano 2014 criteria for assessing FDG-PET/CT in systemic lymphoma (Cheson et al, 2014). Measurable disease according to IPCG criteria will be assessed by brain/spine MRI for CNS disease * Subject must have a tumor biopsy sample (at least 16 unstained slides of tissue or tissue block) from the most recent relapse available prior to MB-CART2019.1 infusion. If medically not feasible to obtain a biopsy from the most recent relapse and for cases when the amount of tissue is limited, the sponsor should be consulted, to confirm adequacy of the sample for study required analyses * No clinical suspicion of central nervous system (CNS) lymphoma (not applicable to CNS cohort) * If the subject has history of CNS disease (not applicable to CNS cohort), then he/she must have no signs or symptoms of CNS disease, have no active disease on magnetic resonance imaging (MRI), have no large cell lymphoma present in cerebral spinal fluid (CSF), regardless of the number of white blood cells (WBCs) * If has history of cerebral vascular accident (CVA), the CVA event must be greater than 12 months prior to leukapheresis. Any neurological deficits must be stable * A creatinine clearance (as estimated by direct urine collection or Cockcroft-Gault Equation) \> 45mL/min * Cardiac ejection fraction (EF) ≥ 45% as determined by an echocardiogram (ECHO) or Multigated Radionuclide Angiography (MUGA) * Resting O2 saturation \>90% on room air * Serum alanine aminotransferase (ALT) / aspartate aminotransferase (AST)\<5 times the Upper Limit of Normal (ULN) for age * Total bilirubin \<1.5 mg/dl, except in individuals with Gilbert's syndrome * Absolute neutrophil count (ANC) \> 1000/μL * Absolute lymphocyte count \> 100/μL * Platelet count \> 50,000/µL * Estimated life expectancy of more than 3 months other than primary disease Exclusion Criteria: * Primary CNS lymphoma (not applicable to CNS cohort) * Richter's transformed DLBCL arising from chronic lymphocytic leukemia (CLL) (not applicable to RT cohort) * Unable to give informed consent * Known history of infection with human immunodeficiency virus (HIV) or active hepatitis B (HBsAg positive). If there is a history of treated hepatitis B or hepatitis C, the viral load must be quantitative polymerase chain reaction (PCR) negative; antiviral prophylaxis is required if HBsAg negative and anti-HBc positive * Known history of infection with hepatitis C virus (anti-HCV positive) unless viral load is undetectable per quantitative PCR and/or nucleic acid testing. * Pharmacologically uncontrolled seizures. * Known history or presence of autoimmune CNS disease, such as multiple sclerosis, optic neuritis, or other immunologic or inflammatory disease * Presence of CNS disorder that, in the judgment of the investigator, may impair the ability to evaluate neurotoxicity. For CNS Cohort: * Midline shift on MRI * Abnormal high CSF opening pressure and or CSF protein \>150 mg/dL Recent (within 3 months) whole brain radiotherapy (WBRT) * Active systemic fungal, viral, or bacterial infection * Pregnant or breast-feeding woman * Previous or concurrent malignancy with the following exceptions: * Adequately treated basal cell or squamous cell carcinoma (adequate wound healing required prior to study entry) * In situ carcinoma of the cervix or breast, treated curatively and without evidence of recurrence for at least 2 years prior to the study * Adequately treated breast or prostate carcinoma on hormonal therapies such as Lupron or tamoxifen and in clinical remission of ≥ 2 years * A primary malignancy which has been completely resected / treated with curative intent and in complete remission of ≥ 2 years * Severely immunocompromised subjects e.g., due to current treatment of non-neurologic autoimmune disease (e.g., Crohn's disease, rheumatoid arthritis, systemic lupus erythematosus). * Medical condition requiring prolonged use of systemic corticosteroids equivalent to prednisone \>10 mg/day. For CNS cohort: Up to 2 mg/day dexamethasone (or equivalence) may be allowed at any time, higher doses allowed up to 7 days prior to apheresis or after apheresis until lymphodepletion. * History of myocardial infarction, cardiac angioplasty or stenting, unstable angina, or other clinically significant cardiac disease within 6 months of enrollment * Concurrent radiotherapy (normal tissue sparing palliative radiotherapy allowed up to time of lymphodepletion). For systemic therapy, at least 2 weeks or 5 half-lives, whichever is shorter, must have elapsed at the time of scheduled leukapheresis. * Baseline dementia that would interfere with therapy or monitoring, determined using Immune Effector Cell-Associated Encephalopathy (ICE) Assessment at baseline * History of severe immediate hypersensitivity reaction to any of the agents used in this study * Refusal to participate in additional lentiviral gene therapy LTFU protocol * Prior CAR-T therapy for any indication or systemic gene modifying therapy for B-cell lymphoma * Prior allogeneic stem cell transplant for any indication * Prior BITE antibodies for cancer therapy * Prior T cell receptor-engineered T cell therapy