This randomized phase III trial studies how well initial treatment with ipilimumab and nivolumab followed by dabrafenib and trametinib works and compares it to initial treatment with dabrafenib and trametinib followed by ipilimumab and nivolumab in treating patients with stage III-IV melanoma that contains a mutation known as BRAFV600 and cannot be removed by surgery. Immunotherapy with monoclonal antibodies, such as ipilimumab and nivolumab, may help the body’s immune system attack the cancer, and may interfere with the ability of tumor cells to grow and spread. Dabrafenib and trametinib may block tumor growth by targeting the BRAFV600 gene. It is not yet known whether treating patients with ipilimumab and nivolumab followed by dabrafenib and trametinib is more effective than treatment with dabrafenib and trametinib followed by ipilimumab and nivolumab.