Cellular Immunotherapy Following Chemotherapy in Treating Patients With Recurrent Non-Hodgkin Lymphomas, Chronic Lymphocytic Leukemia, or B-Cell Prolymphocytic Leukemia

Inclusion Criteria: * SCREENING INCLUSION CRITERIA: * COH pathology review confirms that research participant's diagnostic material is consistent with recurrent/progressive/residual B cell lymphoproliferative neoplasms as listed below AND the research participant is not eligible for or declines COH IRB Protocol No. 13277; additionally, CD19 positivity must be documented in a pathology report if the research participant previously received CD19-targeted therapy; however, it is not a requirement that the CD19 testing be performed by a COH pathologist * Disease stratum 1 (NHL): Unclassifiable high grade lymphoma, mantle cell lymphoma, follicular lymphoma, diffuse large B-cell lymphoma (DLBCL) and all its subtypes, Burkitt lymphoma (BL), marginal zone B-cell lymphoma, hairy cell leukemia, lymphoplasmacytic lymphoma, B cell lymphoma unclassifiable with features intermediate between DLBCL and BL, B cell lymphoma unclassifiable with features intermediate between DLBCL and classical Hodgkin lymphoma, and those research participants who either declined or were not eligible for COH IRB Protocol No. 13277, or who collected autologous T cells for COH IRB Protocol No. 13277 but then became ineligible for autologous hematopoietic stem cell transplant (HSCT) or participants who have relapsed following prior T cell therapy on either COH IRB Protocol No. 09174 or 12224 may be enrolled on this study * Disease stratum 2 (CLL/PLL/SLL): chronic lymphocytic leukemia (CLL), and B-cell prolymphocytic leukemia (PLL), and small lymphocytic lymphoma (SLL) * Karnofsky performance status (KPS) of \>= 70% * Life expectancy \>= 16 weeks at time of screening * The effects of CD19R(EQ)28zeta/EGFRt+ TCM or CD19R(EQ)28zeta/EGFRt+ TN/MEM on the developing fetus are unknown; for this reason, women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control or abstinence) prior to study entry and for six months following duration of study participation; should a woman become pregnant or suspect that she is pregnant while participating on the trial, she should inform her treating physician immediately * All subjects must have the ability to understand and the willingness to sign a written informed consent * Note: For research participants who do not speak English, a short form consent may be used with a COH certified interpreter/translator to proceed with screening and leukapheresis, while the request for a translated full consent is processed; however, the research participant is allowed to proceed with lymphodepletion and T cell infusion only after the translated full consent form is signed * PROTOCOL-SPECIFIC CRITERIA: * COH pathology review confirms that research participant's diagnostic material is consistent with a lymphoproliferative B-cell neoplasm * Documentation of recurrence/progression/residual disease following prior therapy * Negative serum pregnancy test for women of childbearing potential * A pretreatment creatinine clearance (CrCl) of \>= 60 mL/minute (min), calculated by Cockcroft Gault * Patients must have a serum bilirubin =\< 2.0 mg/dl * Patients must have an alanine aminotransferase (ALT) and aspartate aminotransferase (AST) =\< 2.5 times the institutional upper limits of normal * Ejection fraction measured by echocardiogram or multi gated acquisition scan (MUGA) \> 45% (evaluation within 6 weeks of screening does not need to be repeated) * ELIGIBILITY TO PROCEED WITH PERIPHERAL BLOOD MONONUCLEAR CELL (PBMC) COLLECTION * Research participant must have appropriate venous access * Research participant must be at least 2 weeks from having received the last dose of immunosuppressant medications (e.g. calcineurin inhibitors, methotrexate, immunosuppressive antibodies, etc) * The last dose of prior chemotherapy, immunotherapy or radiation must be at least 2 weeks before the leukapheresis procedure * The last dose of cytotoxic chemotherapeutic agents that are not considered lymphotoxic must be at least one week before the leukapheresis procedure; oral chemotherapeutic agents, including lenalidomide and ibrutinib, are allowed if at least 3 half-lives have elapsed prior to leukapheresis * The last dose of lymphotoxic chemotherapeutic agents (e.g. cyclophosphamide, ifosofamide, bendamustine, etc) must be at least 2 weeks before the leukapheresis procedure * The last dose of investigational agents must be at least 2 weeks before leukapheresis procedure unless no response or disease progression is documented on the experimental therapy and at least 3 half-lives must have elapsed prior to leukapheresis * Note: exceptions may be made at the discretion of the principal investigator (PI)/study team * ELIGIBILITY TO UNDERGO LYMPHODEPLETION: * Research participant has a released cryopreserved T cell product for T cell infusions on approximately day 0 * Research participant must be at least 2 weeks out from having received the last dose of investigational agent * The last dose of cytotoxic chemotherapeutic agents that are not considered lymphotoxic must be at least one week before lymphodepletion; oral chemotherapeutic agents, including lenalidomide and ibrutinib, are allowed if at least 3 half-lives have elapsed prior to lymphodepletion * The last dose of lymphotoxic chemotherapeutic agents (e.g. cyclophosphamide, ifosofamide, bendamustine, etc) must be at least 2 weeks before lymphodepletion * Toxicity related to prior therapy must either have returned to =\< grade 3, baseline, or deemed irreversible * KPS \>= 70% * Participants of reproductive potential must agree to use and utilize an adequate method of contraception throughout treatment and for at least 8 weeks after T cell infusion * Absolute neutrophil count (ANC) \> 0.75 * Platelets \> 50 K without growth factor or transfusion support for a week at least * Not requiring supplemental oxygen or mechanical ventilation, oxygen saturation 90% or higher on room air * Not requiring pressor support, not having symptomatic cardiac arrhythmias * Preservation of renal function, serum creatinine did NOT increase by more than 2 fold above the normal range * Total bilirubin =\< 2.0 mg/dL * Research participant without clinically significant encephalopathy/new focal deficits * No clinical evidence of uncontrolled active infectious process * (ELIGIBILITY CRITERIA AT TIME OF INFUSION OF GENETICALLY MODIFIED AUTOLOGOUS T CELLS): Research participant has completed prescribed lymphodepletion * (ELIGIBILITY CRITERIA AT TIME OF INFUSION OF GENETICALLY MODIFIED AUTOLOGOUS T CELLS): Not requiring supplemental oxygen or mechanical ventilation, oxygen saturation 90% or higher on room air * (ELIGIBILITY CRITERIA AT TIME OF INFUSION OF GENETICALLY MODIFIED AUTOLOGOUS T CELLS): Not requiring pressor support, not having symptomatic cardiac arrhythmias * (ELIGIBILITY CRITERIA AT TIME OF INFUSION OF GENETICALLY MODIFIED AUTOLOGOUS T CELLS): Preservation of renal function, serum creatinine did NOT increase by more than 2 fold above the normal range * (ELIGIBILITY CRITERIA AT TIME OF INFUSION OF GENETICALLY MODIFIED AUTOLOGOUS T CELLS): Total bilirubin =\< 2.0 mg/dL * (ELIGIBILITY CRITERIA AT TIME OF INFUSION OF GENETICALLY MODIFIED AUTOLOGOUS T CELLS): Research participant without clinically significant encephalopathy/new focal deficits * (ELIGIBILITY CRITERIA AT TIME OF INFUSION OF GENETICALLY MODIFIED AUTOLOGOUS T CELLS): No clinical evidence of uncontrolled active infectious process Exclusion Criteria: * SCREENING EXCLUSION CRITERIA: * Research participants who received memory-enriched CD19R(EQ):CD28:zeta/EGFRt+ on IRB#13277 * Research participants with any uncontrolled illness including ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, poorly controlled pulmonary disease or psychiatric illness/social situations that would limit compliance with study requirements * Research participants with known active hepatitis B or C infection; research participants who are human immunodeficiency virus (HIV) seropositive based on testing performed within 4 weeks of screening; research participants with any signs of symptoms of active infection, positive blood cultures or radiological evidence of infections * Research participants with presence of other active malignancy; however, research participants with history of prior malignancy treated within 2 years with curative intent and in a complete remission are eligible * Pregnant and lactating women * STUDY-SPECIFIC EXCLUSIONS: * Failure of research participant to understand the basic elements of the protocol and/or the risks/benefits of participating in this phase I study * Research participants with precursor B-cell acute lymphoblastic leukemia/lymphoma or plasma cell dyscrasias * Any known contraindications to cyclophosphamide, fludarabine, etoposide, bendamustine, cetuximab or tocilizumab * Dependence on corticosteroids * Steroid dependence can be defined as a medical need to be greater than 5 mg of prednisone (or equivalent doses of other systemic steroids) a day, chronically; higher doses need to be avoided for at least 3 days prior to leukapheresis and, again, for at least 3 days prior to T cell infusion and up to at least 3 months after T cell infusion unless medically indicated to treat a new toxicity * Note: topical and inhaled corticosteroids in standard doses and physiologic replacement for subjects with adrenal insufficiency are allowed * Active autoimmune disease requiring systemic immunosuppressive therapy * Subjects, who in the opinion of the investigator, may not be able to comply with the safety monitoring requirements of the study