This phase II trial studies how well ixazomib citrate, pomalidomide, dexamethasone, and stem cell transplantation works in treating patients with multiple myeloma that has come back or does not respond to treatment. Giving chemotherapy, such as pomalidomide and dexamethasone, before a stem cell transplant helps kill any cancer cells that are in the body and helps make room in the patient?s bone marrow for new blood-forming cells (stem cells) to grow. After treatment, stem cells are collected from the patient's blood and stored. More chemotherapy is then given to prepare the bone marrow for the stem cell transplant. The stem cells are then returned to the patient to replace the blood-forming cells that were destroyed by the chemotherapy. Giving ixazomib citrate in addition to pomalidomide, dexamethasone, and stem cell transplantation may work better in treating patients with relapsed or refractory multiple myeloma.