Genetically Engineered Cells (MAGE-A1-specific T Cell Receptor-transduced Autologous T-cells) and Atezolizumab for the Treatment of Metastatic Triple Negative Breast Cancer, Urothelial Cancer, or Non-small Cell Lung Cancer

Tissue confirmation of triple negative breast cancer (TNBC), urothelial carcinoma or non-small cell lung cancer (NSCLC) and expression of MAGEA1: Participants must have metastatic disease. Confirmation of diagnosis must be or have been performed by internal pathology review of archival, initial or subsequent biopsy or other pathologic material at Fred Hutchinson Cancer Research Center (FHCRC)/Seattle Cancer Care Alliance (SCCA)/University of Washington Medical Center (UWMC). Patients with TNBC must meet the American Society of Clinical Oncology - College of American Pathologists (ASCO-CAP) definition of negative estrogen, progesterone and HER2 receptor expression. Baseline tissue will be stained to confirm MAGE-A1 expression

Measurable disease by RECIST 1.1 criteria: Participants must have measurable disease, defined as at least one target lesion that can be measured in at least one dimension (longest diameter to be recorded) as \>= 10 mm, unless lymph node in which case short axis must be \>= 15 mm. Baseline imaging (for example diagnostic computed tomography \[CT\] chest/abdomen/pelvis and imaging of the affected extremity as appropriate), brain imaging (magnetic resonance imaging \[MRI\] or CT scan) must be obtained within 45 days of prior to start of first planned FH-MAGEA1-A2TCR infusion. MRI can be substituted for CT in patients unable to have CT contrast. Measurable disease is not required for purposes of leukapheresis /cell manufacturing storage for patients who meet HLA and expression criteria but not standard treatment criteria

Previous treatment with standard of care (SOC) Food and Drug Administration (FDA)-approved therapies. Patients with NSCLC who have actionable somatic mutations or alterations in EGFR, ROS1 and ALK with FDA-approved drug therapy options will be eligible for study only after treatment with targeted therapies for those mutations have been offered or received. Patients with urothelial carcinoma who are candidates for enfortumab vedotin (enfortumab vedotin-ejfv) will be eligible for study after prior treatment with enfortumab vedotin-ejfv has been offered or received

\* Note: Participants will be eligible to enroll before standard therapy is received in order to expedite leukapheresis/cell manufacturing as long as the aforementioned agent is administered prior to lymphodepletion and/or cell infusion

Previous treatment with PD-1 axis inhibitor: Patients in Phase I/2 must have been offered or been previously treated with at least one dose of a PD-L1 axis inhibitor (e.g. PD-1 or PD-L1 inhibiting monoclonal antibody such as pembrolizumab, nivolumab, avelumab, atezolizumab, durvalumab). If received, they must have either developed progression or still have detectable disease and not have developed Common Terminology Criteria for Adverse Events (CTCAE) grade 3 or higher toxicity while on treatment. Patients may have received 1 or more prior systemic regimens for metastatic TNBC or NSCLC. There is no upper limit on prior regimens. Patients may have received prior anti-PD-1/anti-PD-L1 in the neoadjuvant or adjuvant setting

HLA type HLA-A\*02:01: Participants must be HLA-A\*02:01 in order for infused transgenic T cells in order to insure recognition of antigen-MHC complexes. HLA typing should be determined though a molecular approach in a clinical laboratory licensed for HLA typing

Life expectancy must be anticipated to be \> 3 months at trial entry

18 years or older

Capable of understanding and providing a written informed consent

If fertile, willingness to comply with reproductive requirements

Eastern Cooperative Oncology Group (ECOG) performance status of 0, 1, or 2

Tumor tissue amenable to safe biopsy and subject willing to undergo serial tumor biopsies: Should there be no tumor tissue that is accessible for biopsy, patients will still be considered for participation, at discretion of the sponsor and in consultation with the investigator. Similarly, should an investigator determine that a biopsy cannot be performed safely for clinical reasons biopsies may be cancelled or retimed after confirming plan with the sponsor

Participants must be at least three weeks from last systemic treatment at the time of cell collection: At least 3 weeks must have passed since any: immunotherapy (for example, T-cell infusions, immunomodulatory agents, interleukins, vaccines), small molecule or chemotherapy cancer treatment, other investigational agents. There is no washout period for radiation, so long as radiated lesion is not the lesion being evaluated for RECIST measurements on the protocol. Bisphosphonates are permitted but the concurrent treatment with RANK ligand inhibitors (i.e., denosumab) is not permitted within 8 weeks of treatment

Serum creatine \< 2.5 mg/dL or estimated glomerular filtration rate (eGFR) \> 30 mL/min

Total bilirubin (tBili) \< 3.0 mg/dL. Patients with suspected Gilbert syndrome may be included if Tbili \> 3 but no other evidence of hepatic dysfunction

Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) \< 5 x upper limit of normal (ULN)

=\< grade 1 dyspnea and oxygen saturation (SaO2) \>= 92% on ambient air. If pulmonary function tests (PFTs) are performed based on the clinical judgement of the treating physician, patients with forced expiratory volume in 1 second (FEVI) \>= 50% of predicted and carbon monoxide diffusing capability test (DLCO) (corrected) of \>= 40% of predicted will be eligible

Patients 60 years of age or older are required to have left ventricular ejection fraction (LVEF) evaluation performed within 60 days prior to enrollment. LVEF may be established with echocardiogram or multigated acquisition scan (MUGA) scan, and left ejection fraction must be \>= 35%. Cardiac evaluation for other patients is at the discretion of the treating physician

Absolute neutrophil count (ANC) \> 500 cells/ mm\^3

What this study is about