A Study of Mesothelin-Targeted CAR T-Cell Therapy in People With Esophagogastric Cancer or Malignant Peritoneal Mesothelioma

Inclusion Criteria: * Aged ≥18 years * Diagnosis of pathologically/cytologically confirmed EG adenocarcinoma or MPeM * Diagnosis of metastatic or recurrent disease * ECOG performance status of 0-1 * Life expectancy of ≥4 months Inclusion Criteria for Leukapheresis: * Written informed consent for the study (from participant) * Life expectancy of ≥4 months * ECOG performance status of 0-1 * Histologic diagnosis that \& \>25% of the tumor expresses MSLN by IHC analysis. Archival tissue obtained up to 2 years before study enrollment is acceptable. IHC testing of a cell block from cytology (e.g., ascitic fluid) is acceptable if approved by the study pathologist. If adequate archival tissue is not available at screening, a fresh tumor biopsy should be obtained * Stage IV disease with gross peritoneal carcinomatosis on imaging and/or microscopic peritoneal involvement by cytology or noted during diagnostic laparoscopy * Disease progression or treatment intolerance after receiving at least 1 treatment regimen in the metastatic setting; patients with disease recurrence within 6 months of completing curative systemic therapy (chemotherapy, chemoradiation or adjuvant immunotherapy) are also eligible * Patients with Her2 positive disease must have received ≥1 line of anti-Her2 based therapy * At least 1 measurable or evaluable lesion per RECIST 1.1. Screening imaging must be obtained within 6 weeks of signing the informed consent form * Completion of systemic therapy at least 7 days before leukapheresis o Immune checkpoint inhibitor therapy must be completed at least 14 days before leukapheresis * Lab requirements (hematology): * Absolute neutrophil count ≥1.0 K/mcL * Hemoglobin ≥9 gm/dL * Platelet count ≥75 K/mcL * Blood product transfusion or growth factor support cannot occur within 7 days of testing * Lab requirements (serum chemistry): * Bilirubin ≤1.5× upper limit of normal (ULN) * Serum alanine aminotransferase and serum aspartate aminotransferase (ALT/AST) level ≤3× ULN * Calculated clearance of ≥50 mL/min by Cockcroft-Gault equation * Negative screen for infectious disease markers, including hepatitis B core antibody, hepatitis B surface antigen, hepatitis C antibody, HIV 1-2 antibody, HTLV antibody and syphilis antibody o Note: Patients with a history of hepatitis B virus infection are eligible if the hepatitis B viral load is undetectable. Patients with a history of hepatitis C virus infection who were treated for hepatitis C and cured are eligible if the hepatitis C viral load is undetectable * Serum pregnancy test with negative result at screening and preconditioning and must be willing to use effective and reliable contraception for at least 12 months after T cell infusion (for female participants of childbearing age) * Resolution of all acute toxic effects of any previous therapeutic or palliative chemotherapy, radiotherapy, or surgical procedures to grade ≤1 (CTCAE v5.0), except for neuropathy and alopecia Inclusion Criteria for lymphodepleting chemotherapy/CAR T cell infusion * Life expectancy of ≥4 months * ECOG performance status of 0-1 * At least 1 measurable or evaluable lesion per RECIST 1.1. Screening imaging must be obtained within 4 weeks before the date of lymphodepletion * Completion of systemic therapy at least 14 days before lymphodepleting chemotherapy o Immune checkpoint inhibitor therapy must be completed at least 28 days before lymphodepleting chemotherapy * Lab requirements (hematology): * Absolute neutrophil count ≥1.5 K/mcL * Hemoglobin ≥8 gm/dL * Platelet count ≥75 K/mcL * Lab requirements (serum chemistry): * Bilirubin ≤1.5× upper limit of normal (ULN) * Serum alanine aminotransferase and serum aspartate aminotransferase (ALT/AST) level ≤3× ULN * Calculated clearance of ≥50 mL/min by Cockcroft-Gault equation * Serum pregnancy test with negative result within 7 days of planned lymphodepletion date and must be willing to use effective and reliable contraception for at least 12 months after T cell infusion (for female participants of childbearing age) * Resolution of all acute toxic effects of any previous therapeutic or palliative chemotherapy, radiotherapy, or surgical procedures to grade ≤1 (CTCAE v5.0), except for neuropathy and alopecia Participant Exclusion Criteria Exclusion Criteria for Leukapheresis or Lymphodepleting chemotherapy/CAR T cell infusion: Participants are excluded from enrollment if any of the following criteria apply: * Pregnant or lactating * HIV, active hepatitis C virus, or active hepatitis B virus infection, as determined by quantitative PCR (patients who have undergone negative testing prior to leukapheresis do not require repeat testing) * Receiving therapy for concurrent active malignancy * Note: Patients receiving treatment for in situ skin malignancies are not excluded. * Patients with any malignancy diagnosed \>3 years before that is thought to be curatively treated and/or has a low risk of recurrence are eligible. Patients may continue to receive adjuvant therapy at the time of study enrollment (e.g., adjuvant hormonal therapy for curatively treated breast cancer). * Known hematologic malignancy requiring treatment in the preceding 5 years or a known history of lymphoid malignancy * Previous receipt of CAR T cell therapy or any other cellular therapy * Previous mesothelin-directed therapy Any major abdominal surgery (laparotomy with resection of gastrointestinal tract or organ resection) that is completed \<28 days before study enrollment. Patients who have undergone diagnostic laparoscopy can be included in the study without regard to timing * Untreated or active central nervous system (CNS) metastases (progressing or requiring anticonvulsants or corticosteroids for symptomatic control). Patients with a history of treated CNS metastases are eligible if all of the following criteria are met: * Radiographic demonstration of improvement upon completion of CNS-directed thera