This phase II trial studies how well daratumumab, bortezomib, and dexamethasone followed by daratumumab, ixazomib, and dexamethasone in treating patients with multiple myeloma that has come back (relapsed) or does not response to treatment (refractory). Immunotherapy with monoclonal antibodies, such as daratumumab, may help the body's immune system attack the cancer, and may interfere with the ability of tumor cells to grow and spread. Drugs used in chemotherapy, such as dexamethasone, work in different ways to stop the growth of cancer cells, either by killing the cells, by stopping them from dividing, or by stopping them from spreading. Bortezomib and ixazomib may stop the growth of cancer cells by blocking some of the enzymes needed for cell growth. Giving daratumumab, bortezomib, and dexamethasone followed by daratumumab, ixazomib, and dexamethasone may work better and help to control cancer in patients with multiple myeloma.