This phase Ib trial studies the side effects of combination chemotherapy and donor stem cell transplant followed by ixazomib citrate maintenance therapy in treating patients with multiple myeloma that has returned after a period of improvement and is likely to recur (come back), or spread. Giving chemotherapy before a donor peripheral blood stem cell transplant helps stop the growth of cells in the bone marrow, including normal blood-forming cells (stem cells) and cancer cells. When the healthy stem cells from a donor are infused into the patient they may help the patient's bone marrow make stem cells, red blood cells, white blood cells, and platelets. Giving ixazomib citrate after the transplant may improve the overall treatment outcome without causing additional toxicities.