Autologous CD8+ and CD4+ Transgenic T Cells Expressing High Affinity KRASG12V Mutation-Specific T Cell Receptors (FH-A11KRASG12V-TCR) in Treating Patients With Metastatic Solid Tumor Cancers With KRAS G12V Mutations

Inclusion Criteria:

* LEUKAPHERESIS: Diagnosis of metastatic solid tumor
* LEUKAPHERESIS: Tissue confirmation of solid tumor. Confirmation of diagnosis must be or have been performed by internal pathology review of archival biopsy material or other pathologic material at Fred Hutch/University of Washington Cancer Consortium (UWMC)
* LEUKAPHERESIS: HLA-A\*11:01 confirmed through HLA typing at a clinically accredited laboratory
* LEUKAPHERESIS: Previously documented KRASG12V mutation in tumor or plasma cell-free deoxyribonucleic acid (cfDNA) specimens by polymerase chain reaction (PCR) or next-generation sequencing (NGS) test
* LEUKAPHERESIS: 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 CT chest/abdomen/pelvis and imaging of the affected extremity as appropriate), brain imaging (MRI or CT scan) if clinically indicated, must be obtained within 8 weeks of the first planned T cell infusion. MRI can be substituted for CT in participants unable to have CT contrast
* LEUKAPHERESIS: Participants must be willing to undergo tumor biopsy for research purposes if safe and feasible at baseline (prior to first T cell infusion), 2-3 weeks after the first T cell infusion, and approximately 2 weeks +/- 1 week after the 2nd infusion (if applicable), if safe and feasible (these time windows may vary due to manufacturing or clinical reasons). Should there be no tumor tissue that is accessible for biopsy, participants will still be considered for participation, at the discretion of the investigator. Similarly, should an investigator determine that a biopsy cannot be performed safely for clinical reasons, biopsies may be cancelled or rescheduled. Patients can also refuse biopsies at any time after enrollment
* LEUKAPHERESIS: Participants must be at least two weeks or five half-lives (for small molecules) from last systemic treatment: At least 2 weeks must have passed since any: immunotherapy (for example, T-cell infusions, immunomodulatory agents, interleukins, vaccines), or chemotherapy cancer treatment. At least five half-lives must have passed from treatment with small molecules or 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 and denosumab are permitted
* LEUKAPHERESIS: Participants must have progressed on or be intolerant to at least one lines of prior therapy including any targeted therapies indicated for participants with each of the tumor types eligible to enroll, as applicable
* LEUKAPHERESIS: Patients with solid tumors harboring targetable molecular alterations including but not limited to EGFR mutations, ALK, ROS, NTRK fusions, microsatellite instability (MSI)-high, tumor mutational burden (TMB) high, BRAF v600 mutations, HER2 amplifications, must have been treated or refused treatment with applicable targeted therapies, as applicable
* LEUKAPHERESIS: Fertile male and female participants must be willing to use an effective contraceptive method before, during, and for at least 4 months after the last A11KRASG12V-TCR infusion
* LEUKAPHERESIS: 18 years or older at the time of enrollment
* LEUKAPHERESIS: Capable of understanding and providing a written informed consent
* LEUKAPHERESIS: Eastern Cooperative Oncology Group (ECOG) performance status of =\< 1

  * No uncontrolled pleural effusion, pericardial effusion, or ascites requiring repeated drainage more than once every 28 days
* LEUKAPHERESIS: No uncontrolled pleural effusion, pericardial effusion, or ascites requiring repeated drainage more than once every 28 days
* LEUKAPHERESIS: Renal: Creatinine clearance \>= 50 ml/min by Chronic Kidney Disease Epidemiology Collaboration (CKD-EPI) or 24-hour urine clearance
* LEUKAPHERESIS: Hepatic: Total bilirubin \< 2.0 mg/dL
* LEUKAPHERESIS: Hepatic: Aspartate transferase (AST) and alanine transaminase (ALT) \< 5x upper limit of normal (ULN)
* LEUKAPHERESIS: Hepatic: Participants with suspected Gilbert syndrome may be included if total bilirubin (Bili) \> 3 mg/dL but no other evidence of hepatic dysfunction
* LEUKAPHERESIS: Cardiac: Participants 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 MUGA scan, and LVEF must be \>= 35%. Cardiac evaluation for other participants is at the discretion of the treating physician
* LEUKAPHERESIS: Hematologic: Absolute neutrophil count (ANC) \>= 1000 cells/ mm\^3
* LEUKAPHERESIS: Nutrition: Albumin \>= 3 g/dL
* START OF TREATMENT: 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 CT chest/abdomen/pelvis and imaging of the affected extremity as appropriate), brain imaging (MRI or CT scan) must be obtained within 8 weeks of the first planned T cell infusion. MRI can be substituted for CT in participants unable to have CT contrast
* START OF TREATMENT: Participants must be willing to undergo tumor biopsy for research purposes if safe and feasible at baseline (prior to first T cell infusion), 2-3 weeks after the first T cell infusion, and approximately 2 weeks +/- 1 week after the 2nd infusion (if applicable), if safe and feasible (these windows may vary due to manufacturing or clinical reasons). Should there be no tumor tissue that is accessible for biopsy, participants will still be considered for participation, at discretion of the investigator. Similarly, should an investigator determine that a biopsy cannot be performed safely for clinical reasons, biopsies may be cancelled or rescheduled
* START OF TREATMENT: Participants must be at least two weeks from last systemic treatment: At least 2 weeks must have passed since any: immunotherapy (for example, T-cell infusions, immunomodulatory agents, interleukins, vaccines), or chemotherapy cancer treatment. At least five half-lives must have passed from treatment with small molecules or 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 and denosumab are permitted
* START OF TREATMENT: ECOG performance status of =\< 1
* START OF TREATMENT: Renal: Creatinine clearance \>= 50 ml/min by CKD-EPI or 24-hour urine clearance

  * Exceptions may be at the discretion of the PI to allow participants with organ function affected by their organ-specific tumor involvement
* START OF TREATMENT: Hepatic: Total bilirubin \< 2.0 mg/dL

  * Exceptions may be at the discretion of the PI to allow participants with organ function affected by their organ-specific tumor involvement
* START OF TREATMENT: AST and ALT \< 5x upper limit of normal (ULN)
* START OF TREATMENT: Participants with suspected Gilbert syndrome may be included if total Bili \> 3 mg/dl but no other evidence of hepatic dysfunction
* START OF TREATMENT: Pulmonary: =\< grade 1 dyspnea and oxygen saturation of arterial blood (SaO2) \>= 92% on ambient air. If pulmonary function tests (PFTs) are performed based on the clinical judgement of the treating physician, participants with forced expiratory volume in the first second (FEVI) \>= 50% of predicted and diffusing capacity of the lungs for carbon monoxide (DLCO) (corrected) \>= 40% of predicted will be eligible

  * Exceptions may be at the discretion of the PI to allow participants with organ function affected by their organ-specific tumor involvement
* START OF TREATMENT: Hematologic: ANC \>= 1000 cells/ mm\^3

  * Exceptions may be at the discretion of the PI to allow participants with organ function affected by their organ-specific tumor involvement
* START OF TREATMENT: Nutrition: Albumin \>= 3 g/dL

  * Exceptions may be at the discretion of the PI to allow participants with organ function affected by their organ-specific tumor involvement
* START OF TREATMENT: Participants with a history of chronic obstructive pulmonary disease (COPD), emphysema, or greater than 30 pack year smoking history should undergo PFTs and meet the following criteria:

  * FEVI \>= 50% of predicted and DLCO (corrected) \>= 40% of predicted will be eligible

Exclusion Criteria:

* LEUKAPHERESIS: Prior solid organ transplant or allogeneic hematopoietic stem cell transplant: Kidney transplant participants will be considered on a case-by-case basis requiring discussion with PI. If the participant has had a kidney transplant, participant must have dialysis access, dialysis plan, supportive nephrologist, willingness to stop transplant immunosuppression, and express understanding that rejection is possible outcome. Dialysis or costs related to transplant kidney will not be supported by the study. Participants having had any other solid organ transplants will be excluded, as will those with any history of allogeneic hematopoietic stem cell transplant
* START OF TREATMENT: Pregnancy, breastfeeding, or expecting to conceive or father children for the duration of the trial through 4 months after last T-cell infusion. Participants of childbearing potential must have a negative serum pregnancy test within the 2 weeks (14 days) preceding T cell infusion. Childbearing potential is defined as women who have not been surgically sterilized and who are not post-menopausal (free of menses for at least 1 year)
* START OF TREATMENT: Active autoimmune disease: Participants with active autoimmune disease requiring immunosuppressive therapy are excluded. Case by case exemptions are possible with approval by PI
* START OF TREATMENT: Corticosteroid therapy at a dose equivalent of \> 0.5 mg/kg of prednisone per day. The following treatments are permitted: intranasal, inhaled, topical, or local steroid applications; systemic corticosteroids at physiologic doses equivalent to no more than \> 0.5 mg/kg/day prednisone; steroids as premedication for contrast dye allergy
* START OF TREATMENT: Concurrent use of other investigational anti-cancer agents
* START OF TREATMENT: Active uncontrolled infection: Human immunodeficiency virus (HIV) positive participants on highly active antiretroviral therapy (HAART) with a CD4 count \> 500 cells/mm\^3 are considered controlled, as are individuals with a history of hepatitis C who have successfully completed antiviral therapy with an undetectable viral load, and those with hepatitis B who have hepatitis well controlled on medication
* START OF TREATMENT: Uncontrolled concurrent illness: Participants may not have uncontrolled or concurrent illness including, but not limited to, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements
* START OF TREATMENT: Untreated brain metastases: Participants with small asymptomatic brain metastases (\< 1 cm) or those with brain metastases previously treated and controlled with surgery or radiotherapy will be considered for inclusion at discretion of PI, so long as all other eligibility criteria are met
* START OF TREATMENT: Active treatment for prior immune related adverse event to any immunotherapy: Participants receiving ongoing treatment for prior serious immune-related adverse events are excluded, with exception of hormone supplementation or corticosteroid therapy at equivalent of up to 0.5mg/kg/day prednisone per day, unless otherwise approved by PI
* START OF TREATMENT: Other medical, social, or psychiatric factor that interferes with medical appropriateness and/or ability to comply with study, as determined by the PI
* START OF TREATMENT: Known allergic reactions to any of the components of study treatments
* START OF TREATMENT: Uncontrolled pleural effusion, pericardial effusion, or ascites requiring repeated drainage more than once every 28 days