Administering Peripheral Blood Lymphocytes Transduced With a Murine T-Cell Receptor Recognizing the G12D Variant of Mutated RAS in HLA-A*11:01 Patients

-INCLUSION CRITERIA: 1. Measurable (per Response Evaluation Criteria in Solid Tumors v1.1 criteria), metastatic, or unresectable malignancy expressing G12D mutated KRAS as assessed by one of the following methods: Reverse Transcription Polymerase Chain Reaction (RT-PCR) on tumor tissue, tumor deoxyribonucleic acid (DNA) sequencing, or any other Clinical Laboratory Improvement Amendments (CLIA)-certified laboratory test on resected tissue. Patients shown to have tumors expressing Glycine(G) to Aspartic Acid(D) substitution at codon 12(G12D) mutated neuroblastoma rat sarcoma (NRAS) and Harvey rat sarcoma viral oncogene homolog (HRAS) will also be eligible as these oncogenes share complete amino acid homology with G12D mutated KRAS for their first 80 N-terminal amino acids, completely encompassing the target epitope. 2. Patients must be human leukocyte antigens (HLA) -A\*11:01 positive as confirmed by the National Institutes of Health (NIH) Department of Transfusion Medicine. 3. Confirmation of the diagnosis of cancer by the National Cancer Institute (NCI) Laboratory of Pathology. 4. Patients must have: -previously received standard systemic therapy for their advanced cancer and have been either non-responders or have recurred, specifically: * Patients with metastatic colorectal cancer must have had at least two systemic chemotherapy regimens that include 5-Fluorouracil (5-FU), leucovorin, bevacizumab, oxaliplatin, and irinotecan (or similar agents), or have contraindications to receiving those medications. * Patients with pancreatic cancer must have received gemcitabine, 5FU, and oxaliplatin (or similar agents), or have contraindications to receiving those medications. * Patients with non-small cell lung cancer (NSCLC) must have had appropriate targeted therapy as indicated by abnormalities in anaplastic lymphoma kinase (ALK), estimated glomerular filtration rate (EGFR), or expression of programmed death-ligand 1 (PD-L1). Other patients must have had platinum-based chemotherapy. * Patients with ovarian cancer or prostate cancer must have had approved first-line chemotherapy. OR -declined standard treatment. 5. Patients with 3 or fewer brain metastases that are \< 1 cm in diameter and asymptomatic are eligible. Lesions that have been treated with stereotactic radiosurgery must be clinically stable for one month after treatment for the patient to be eligible. Patients with surgically resected brain metastases are eligible. 6. Age greater than or equal to 18 years and less than or equal to 72 years. 7. Clinical performance status of Eastern Cooperative Oncology Group (ECOG) 0 or 1 8. Patients must be willing to practice birth control from the time of enrollment on this study and for 12 months after the last dose of combined chemotherapy for women and for four months after treatment for men. 9. Women of child-bearing potential must be willing to undergo a pregnancy test prior to the start of treatment because of the potentially dangerous effects of the treatment on the fetus. NOTE: Certain malignancies may secrete hormones that produce false positive pregnancy tests. Serial blood testing (e.g. Human chorionic gonadotropin (hCG) measurements) and/ or ultrasound may be performed for clarification. 10. Serology -Seronegative for human immunodeficiency virus (HIV) antibody. (The experimental treatment being evaluated in this protocol depends on an intact immune system. Patients who are HIV seropositive may have decreased immune-competence and thus may be less responsive to the experimental treatment and more susceptible to its toxicities.) -Seronegative for hepatitis B antigen, and seronegative for hepatitis C antibody. If hepatitis C antibody test is positive, then patient must be tested for the presence of antigen by RT-PCR and be hepatitis C virus ribonucleic acid (HCV RNA) negative 11. Hematology * Absolute neutrophil count (ANC) \> 1000/mm\^3 without the support of filgrastim * White blood cell (WBC) greater than or equal to 2500/mm\^3 * Platelet count greater than or equal to 80,000/mm\^3 * Hemoglobin \> 8.0 g/dL. Subjects may be transfused to reach this cut-off. 12. Chemistry * Serum alanine Aminotransferase (ALT)/aspartate aminotransferase (AST) less than or equal to 5.0 x ULN * Serum creatinine less than or equal to 1.6 mg/dL * Total bilirubin less than or equal to 2.0 mg/dL, except in patients with Gilbert's Syndrome, who must have a total bilirubin \< 3.0 mg/dL. 13. Patients must have completed any prior systemic therapy at the time of enrollment. Note: Patients may have undergone minor surgical procedures or limited field radiotherapy within the four weeks prior to enrollment, as long as related major organ toxicities have recovered to less than or equal to grade 1. 14. Ability of subject to understand and the willingness to sign a written informed consent document. 15. Willing to sign a durable power of attorney. 16. Subjects must be co-enrolled on the protocol 03C0277. EXCLUSION CRITERIA: 1. Women of child-bearing potential who are pregnant or breastfeeding because of the potentially dangerous effects of the treatment on the fetus or infant. 2. Concurrent systemic steroid therapy. 3. Active systemic infections requiring anti-infective treatment, coagulation disorders, or any other active or uncompensated major medical illnesses. 4. Any form of primary immunodeficiency (such as Severe Combined Immunodeficiency Disease). 5. Concurrent opportunistic infections (The experimental treatment being evaluated in this protocol depends on an intact immune system. Patients who have decreased immune competence may be less responsive to the experimental treatment and more susceptible to its toxicities.) 6. History of severe immediate hypersensitivity reaction to cyclophosphamide, fludarabine, or aldesleukin. 7. History of coronary revascularization or ischemic symptoms. 8. For select patients with a clinical history prompting cardiac evaluation: last known left ventricular ejection fraction (LVEF) less than or equal to 45%. I. For select patients with a clinical history prompting pulmonary evaluation: known forced expiratory volume at one second (FEV1) less than or equal to 50%. j. Patients who are receiving any other investigational agents.