CD19 CAR T Cells for B Cell Malignancies After Allogeneic Transplant

Inclusion Criteria: * Patients with CD19+ B cell malignancy who have persistent, relapsed or progressive disease after hematopoietic stem cell transplant from an human leukocyte antigen (HLA)-matched related donor OR patients with CD19+ B cell malignancy who are planned for or have had a hematopoietic stem cell transplant from an HLA-matched related donor and are at risk of relapse after HCT defined by any one of the disease-specific criteria listed below: * Philadelphia chromosome negative acute lymphoblastic leukemia: * Beyond first complete remission (CR) at the time of pre-transplant evaluation * Required \> 1 cycle of induction chemotherapy to achieve CR * First morphologic CR but with evidence of minimal residual disease by flow cytometry, conventional cytogenetics, fluorescence in situ hybridization (FISH) or polymerase chain reaction (PCR) * First CR with poor risk cytogenetics (t(4:11), t(8;14), hypodiploidy, near triploidy or \> 5 cytogenetic abnormalities) at diagnosis * Planned for or have had a reduced intensity conditioned or non-myeloablative transplant * Philadelphia positive acute lymphoblastic leukemia * Not in CR at the time of pre-transplant evaluation * In CR with the following features: * Intolerant or unwilling to use a TKI after HCT * Current or previous detection of cytogenetic abnormalities in addition to t(9;22) by conventional karyotyping, FISH or molecular methods * Chronic lymphocytic leukemia, or low grade B cell lymphomas: * Failed or ineligible for prior immunochemotherapy that included a purine analog and anti-CD20 monoclonal antibody AND a lymph node \>= 5 cm at the time of pre-transplant evaluation * Mantle cell lymphoma: * Failed or ineligible for autologous transplant AND a lymph node \>= 2 cm at the time of pre-transplant evaluation * Diffuse large B cell lymphomas, large B cell transformation of an indolent lymphoma or other aggressive B cell lymphomas * Failed or ineligible for autologous transplant AND not in CR at the time of pre-transplant evaluation * Confirmation of tumor diagnosis and expression of CD19 after review by University of Washington Medical Center (UWMC) or Seattle Cancer Care Alliance (SCCA) pathology services * The patient has signed the informed consent form for this study * DONOR: Genotypic or phenotypic HLA-identical family members * DONOR: Express one or more of the following combinations of viral serostatus and HLA allele: * CMV seropositive and HLA-A\*0101 positive * CMV seropositive and HLA-A\*0201 positive * CMV seropositive and HLA-B\*0702 positive * CMV seropositive and HLA-B\*0801 positive * EBV seropositive and HLA-A\*0201 positive * EBV seropositive and HLA-B\*0801 positive * DONOR: Hematocrit \>= 35% at enrollment * DONOR: Age \>= 18 years * DONOR: The donor has signed the informed consent form for the study Exclusion Criteria: * Known central nervous system (CNS) tumor (CNS2 or CNS3) that is refractory to intrathecal chemotherapy and/or cranio-spinal radiation; patients with a history of CNS disease that has been effectively treated to CNS1 or lower evidence of disease will be eligible * Human immunodeficiency virus (HIV) seropositive * Significant medical or psychological conditions that would make them unsuitable candidates for T cell therapy * Fertile patients unwilling to use contraception during and for 12 months after protocol enrollment * Pregnant or breast-feeding * DONOR: G-CSF administered within one month prior to the blood draw for T cell collection * DONOR: Unable for any reason to provide a 400 ml blood draw * DONOR: Inadequate peripheral veins for blood collection * DONOR: HIV-1, HIV-2, human T-lymphotropic virus (HTLV)-1 or HTLV-2 seropositive * DONOR: Active hepatitis B or hepatitis C virus infection * DONOR: Positive serologic test for syphilis * DONOR: Aberrant CD45RA isoform expression on all T cells * DONOR: Systolic blood pressure (BP) \< 80 or \> 200 * DONOR: Heart rate \< 50 or \> 120, if considered due to cardiac disease * DONOR: Oxygen (O2) saturation \< 88% on room air * DONOR: Serum creatinine (Cr) \> 3.0 * DONOR: Aspartate aminotransferase (AST) or alanine aminotransferase (ALT) \> 4 x the upper limit of normal * DONOR: Unable to provide informed consent to participate * DONOR: Significant medical conditions (e.g. immunosuppressive therapy) that would make them unsuitable T cell donors * DONOR: Pregnant or nursing