Tacrolimus, Nivolumab, and Ipilimumab in Treating Kidney Transplant Recipients With Selected Unresectable or Metastatic Cancers

Inclusion Criteria: * Patients must be kidney transplant recipients with a functioning allograft who do not currently require dialysis * Patients must have histologically or cytologically confirmed melanoma, basal cell carcinoma, Merkel cell carcinoma, cutaneous squamous cell carcinoma, or microsatellite instability (MSI)-high cancers for which standard non-immunological medical, surgical, or radiation therapy would be insufficient (i.e., patients who are not surgical candidates). This trial is not intended to provide therapy as a neoadjuvant approach * Measurable disease as defined by Response Evaluation Criteria in Solid Tumors (RECIST) 1.1 criteria, i.e., at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded for non-nodal lesions and short axis for nodal lesions) as \>= 20 mm by chest x-ray or as \>= 10 mm with computed tomography (CT) scan, magnetic resonance imaging (MRI), or calipers by clinical exam is preferred, but not required. Patients with evaluable disease but no target lesions (e.g., evaluable bone metastases) may be included after discussion with the principal investigator (PI) * Patients must have documentation, in consultation with the PI, that they received, refused, or were ineligible for the following non-immunologic therapies: * For patients with: * BRAF-mutant melanoma: prior therapies include BRAF/MEK inhibitors * Merkel cell carcinoma: prior therapies include platinum + VP-16 * B…