Brief Summary: This phase II trial studies how well capmatinib, ceritinib, regorafenib, or entrectinib work in treating patients with BRAF/NRAS wild-type stage III-IV melanoma. Capmatinib, ceritinib, regorafenib, or entrectinib may stop the growth of tumor cells by blocking some of the enzymes needed for cell growth.

Detailed Description: PRIMARY OBJECTIVES: I. To estimate the clinical activity of tyrosine kinase inhibitors matched to the tumor-specific fusion kinase in patients with metastatic melanoma. SECONDARY OBJECTIVES: I. To estimate tumor stability in melanoma patients treated with kinase inhibitors matched to the tumor-specific fusion kinase. II. To estimate survival in melanoma patients treated with kinase inhibitors matched to the tumor-specific fusion kinase. III. To examine the safety and tolerability of kinase inhibitors in patients with melanoma with a fusion kinase. TERTIARY OBJECTIVES: I. To explore molecular mechanisms of resistance for patients who progress on therapy. OUTLINE: Patients are assigned to 1 of 4 arms. ARM A: Patients with MET fusion receive capmatinib orally (PO) twice daily (BID) on day 1-28. Courses repeat every 28 days in the absence of disease progression or unaccepted toxicity. ARM B: Patients with ALK fusion receive ceritinib PO once daily (QD) on days 1-28. Courses repeat every 28 days in the absence of disease progression or unaccepted toxicity. ARM C: Patients with RET or BRAF fusion receive regorafenib PO QD on day 1-21. Courses repeat every 28 days in the absence of disease progression or unaccepted toxicity. ARM D: Patients with NTRK1, NTRK2, NTRK3, or ROS1 fusion receive entrectinib PO QD on days 1-28. Courses repeat every 28 days in the absence of disease progression or unaccepted toxicity. After completion of study treatment, patients are followed up within 30 days and then periodically.