Personalized Antibody-Drug Conjugate Therapy Based on RNA and Protein Testing for the Treatment of Advanced or Metastatic Solid Tumors (The ADC MATCH Screening and Treatment Trial)

Inclusion Criteria: * SCREENING PROTOCOL INCLUSION CRITERIA: * Patients must have histologically confirmed solid tumor requiring therapy and meet one of the following criteria: * Patients must have disease not amenable to curative-intent therapy AND * Patients who have had disease progression after treatment with all available therapies for their disease that are known to confer benefit or are intolerant to such treatment will be eligible, if other eligibility criteria are met. If the patient is currently receiving therapy without progression, the clinician must have assessed that the current therapy is no longer benefitting the patient, or that the patient is not tolerating the therapy. Patients can be screened on ADC MATCH if they have had three or fewer-lines of chemotherapy in the advanced/metastatic setting and are expected to need a treatment change within 6 months, and ADC MATCH is felt to be appropriate next line therapy AND * Patients with disease for which no standard treatment exists that has been shown to confer benefit OR * Patients who are willing to forego standard therapies known to confer benefit * NOTE: Patients can be on therapy at the time of initiating the Screening Protocol if the patient is interested in treatment on ADC MATCH upon progression, and the physician deems this appropriate * Patient must have undergone RNA testing in a Clinical Laboratory Improvement Act (CLIA) environment or must submit archival tissue to determine RNA overexpression of ADC TOIs by the TARGET Assay. Patients who have high TOI RNA expression will have confirmation of TOI expression by CLIA IHC assay at MD Anderson Cancer Center (MDACC). Only patients with confirmed TOI protein expression will be eligible for assignment to a treatment cohort. Retrospective confirmation in another central laboratory may also be performed. (A proportion of specimens in patients who are negative for RNA overexpression may be tested to better understand the concordance between the TARGET Assay and immunohistochemistry (IHC) assays, as funding and tissue availability allows. Only patients with RNA and protein expression with be eligible for treatment on ADC MATCH) * Patients must be willing to undergo mandatory pre-treatment and on-treatment tumor biopsies. Patients who do not consent to these research biopsies will not be eligible for prescreening. Patients who have screened and consented to a treatment cohort but are found to have disease that cannot be safely biopsied will be eligible for treatment provided all other eligibility criteria are met * Patients must have measurable disease. (Per radiologic imaging within the past 2 months). * Note: Patients with active metastatic disease that is not measurable but who are expected to be eligible for a Treatment Cohort can be screened after discussion with the primary investigator (PI) * Age ≥ 18 years. Because no dosing or adverse event (AE) data are currently available on the use of the Cancer Therapy Evaluation Program (CTEP) investigational new drug (IND) agents to be used in the study in patients \< 18 years of age, children are excluded from this study * Eastern Cooperative Oncology Group performance status of 0-2 based on most recent assessment (Karnofsky ≥ 50%) * No history of transfusion dependence * No history of persistent bone marrow suppression (absolute neutrophil count ≥ 1,500/mL and platelets ≥ 100,000/mL not attributable to active therapy; patients currently on bone marrow suppressive therapy can undergo assessment for the screening protocol but cannot be treated on any of the treatment cohorts unless bone marrow suppression is reversed off the suppressive therapy) (New testing will not be performed for the screening protocol, but rather existing labs will be assessed to assess eligibility) * Total bilirubin ≤ 1.5 institutional upper limit of normal (ULN). Documented Gilbert syndrome is allowed if total bilirubin is ≤ 3 × ULN (New testing will not be performed for the screening protocol, but rather existing labs will be assessed to assess eligibility) * Aspartate transaminase (serum glutamic-oxaloacetic transaminase \[SGOT\])/alanine transaminase (serum glutamate pyruvate transaminase \[SGPT\]) ≤ 2.5 × institutional ULN. Transaminases up to 5 × ULN in the presence of liver metastases are not allowed to initiate the screening protocol but are allowed for the treatment cohorts (New testing will not be performed for the screening protocol, but rather existing labs will be assessed to assess eligibility) * Creatinine ≤ institutional ULN OR glomerular filtration rate ≥ 60 mL/min/1.73 m\^2 for patients with creatinine levels above institutional normal unless data exists supporting safe use at lower kidney function values, no lower than 30 mL/min/1.73 m\^2 (New testing will not be performed for the screening protocol, but rather existing labs will be assessed to assess eligibility) * Patients must have albumin ≥ 3 g/dL * Patients with a prior or concurrent malignancy whose natural history or treatment does not have the potential to interfere with the safety or efficacy assessment of the investigational agents are eligible for this study * Patients with known history or current symptoms of cardiac disease, or history of treatment with cardiotoxic agents, should have a clinical risk assessment of cardiac function using the New York Heart Association Functional Classification. To be eligible for this study, patients should be class 2B or better * Women of childbearing potential must not be known to be pregnant. Pregnancy testing is not required * Ability to understand and the willingness to sign a written informed consent document * ADDITIONAL INCLUSION CRITERIA FOR TREATMENT COHORTS: * Women of childbearing potential must have a negative serum pregnancy test result at treatment cohort screening * The effects of the study drugs on the developing human fetus are unknown. For this reason and because investigational agents as well as other