CD19-Specific T Cells Post AlloSCT

Inclusion Criteria: * RECIPIENT: Patients with high risk or relapsed disease who are planning to receive, or have received prior allogeneic HSCT from an human leukocyte antigen (HLA)-matched related, or HLA-mismatched related donor; high risk is defined as patients with acute lymphoblastic leukemia who have delayed clearance of minimal residual disease, Philadelphia (Ph)-like, or complex, 11q23 or hypodiploid karyotype * RECIPIENT: Available donor who provided hematopoietic stem-cell (HSC) * RECIPIENT: Patients with CD19+ lymphoid malignancies that are refractory to or intolerant of standard treatment (as defined below): * B-cell Acute Lymphoblastic Leukemia (ALL) * Non-Hodgkin lymphoma (NHL) to include diffuse large B-cell lymphoma (DLBCL) not otherwise specified, primary mediastinal large B cell lymphoma, mantle cell lymphoma, or transformed follicular lymphoma (TFL) as defined by the World Health Organization 2008 criteria * Small lymphocytic lymphoma (SLL) * Chronic lymphocytic leukemia (CLL) * NOTE: Refractory disease for acute and chronic leukemia is defined by: * Presence of \> 5% malignant blasts in bone marrow and/or peripheral blood and/or minimal residual disease by flow cytometry or molecular analysis for fusion proteins and/or positive imaging for extra-medullary disease to most recent therapy * NOTE: Refractory disease for lymphoma is defined as: * Progressive disease or stable disease lasting =\< 6 months, as best response to most recent chemotherapy regimen; or disease progression or recurrence =\< 12 months after prior ASCT * Prior therapy must have included an anti-CD20 monoclonal antibody-containing regimen and an anthracycline-containing chemotherapy regimen * For patients with TFL, prior chemotherapy for follicular lymphoma and subsequent refractory disease after transformation to DLBCL * At least one measurable lesion according to revised International Working Group (IWG) Response Criteria * RECIPIENT: In patients with prior transplant, treatment will begin no earlier than 3 months post-transplant. Enrollment can occur earlier to allow time for donor cell collection * RECIPIENT: Karnofsky performance scale \> 60 * RECIPIENT: Patient able to provide written informed consent * RECIPIENT: Patient able to provide written informed consent for the long-term follow-up (LTFU) gene therapy study * RECIPIENT: Negative human anti-mouse antibody (HAMA) * DONOR: HLA-matched related, HLA-mismatched related, including haploidentical donor, or related donor cleared to donate based on Stem Cell Transplantation and Cellular Therapy (SCTCT) standard-of-care (SOC) guidelines * DONOR: Negative beta HCG in female of child-bearing potential defined as: * Not post-menopausal for 12 months, or * No previous surgical sterilization, or * Lactating females * T-CELL INFUSION REQUIREMENTS FOR RECIPIENT: Patient must have measurable disease at the time of treatment * T-CELL INFUSION REQUIREMENTS FOR RECIPIENT: Not received anti-thymocyte globulin (ATG), Campath, or other T-cell immunosuppressive antibodies or donor-lymphocyte infusion in the 28 days prior to T-cell infusion * T-CELL INFUSION REQUIREMENTS FOR RECIPIENT: Serum creatinine \< 2 x upper limit of normal (ULN) * T-CELL INFUSION REQUIREMENTS FOR RECIPIENT: Alanine aminotransferase (ALT)/aspartate aminotransferase (AST) =\< 2.5 x ULN or =\< 5 x ULN if documented liver metastases * T-CELL INFUSION REQUIREMENTS FOR RECIPIENT: Total bilirubin =\< 1.5 mg/dL, except in patients with Gilbert's syndrome in whom total bilirubin must be =\< 3.0 mg/dL * T-CELL INFUSION REQUIREMENTS FOR RECIPIENT: Cardiac ejection fraction \>= 40%, and no clinically-significant electrocardiogram (ECG) findings * T-CELL INFUSION REQUIREMENTS FOR RECIPIENT: No clinically significant pleural effusion, baseline oxygen saturation \> 90% on room air * T-CELL INFUSION REQUIREMENTS FOR RECIPIENT: No evidence of grade \>= 2 active graft versus host disease (GVHD) using the Center for International Blood and Marrow Transplant Research (CIBMTR) Acute GVHD Grading System or requiring systemic steroid therapy greater than physiologic dosing at time of starting treatment * T-CELL INFUSION REQUIREMENTS FOR RECIPIENT: Non-hematologic toxicity grade \>= 2 (Common Terminology Criteria for Adverse Events \[CTCAE\] version 5) related to the lymphodepleting chemotherapy until the toxicity has resolved to grade =\< 1 and the patient is afebrile * T-CELL INFUSION REQUIREMENTS FOR RECIPIENT: No new grade \> 2 neurologic, pulmonary, cardiac, gastrointestinal, renal or hepatic (excluding albumin) toxicity * T-CELL INFUSION REQUIREMENTS FOR RECIPIENT: Serum creatinine \< 2 x ULN * T-CELL INFUSION REQUIREMENTS FOR RECIPIENT: Oxygen saturation \> 90% on room air * T-CELL INFUSION REQUIREMENTS FOR RECIPIENT: Active clinically significant infection within 7-days of study treatment * T-CELL INFUSION REQUIREMENTS FOR RECIPIENT: Using an investigational agent Exclusion Criteria: * RECIPIENT: Positive beta human chorionic gonadotropin (HCG) in female of child-bearing potential defined as not post-menopausal for 12 months or no previous surgical sterilization or lactating females * RECIPIENT: Patients with allergy to mouse products or cetuximab * RECIPIENT: Active central nervous system (CNS) disease in patient with history of CNS malignancy * RECIPIENT: Positive serology for human immunodeficiency virus (HIV) * RECIPIENT: Active hepatitis B or active hepatitis C * RECIPIENT: Has received a T-cell product within 6 weeks prior to planned infusion of genetically modified T cells