This study will determine the MTD of KK-LC-1 TCR-T cells for the treatment of metastatic cancers that express KK-LC-1 and will assess clinical tumor response to treatment. KK-LC-1 is an abnormal protein that is expressed by certain types of cancer. The cancers that most commonly express KK-LC-1 include gastric, lung, breast, and cervix cancer. KK-LC-1 TCR-T cells are autologous T cells that are genetically engineered to target the KK-LC-1 antigen. Tumor targeting by the KK-LC-1 TCR requires that subjects have the the HLA-A*01:01 allele. Participants will receive a conditioning regimen consisting of cyclophosphamide and fludarabine. They will then receive a single infusion of KK-LC-1 TCR-T cells, which will be followed by administration of aldesleukin. Adverse events, dose limiting toxicity, tumor response, and tumor response duration will be determined.