The goal of this phase II clinical study is to learn about the safety of inotuzumab ozogamicin when given with fludarabine, with or without bendamustine, melphalan, and rituximab before and after a stem cell transplant. Researchers also want to learn if inotuzumab ozogamicin when given after a stem cell transplant can help control leukemia and lymphoma. Inotuzumab ozogamicin is a monoclonal antibody, called inotuzumab, linked to a chemotherapy drug called ozogamicin. Inotuzumab attaches to CD22-positive cancer cells in a targeted way and delivers ozogamicin to kill them. Giving chemotherapy before a bone marrow or 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. Sometimes the transplanted cells from a donor attack the body's normal cells (called graft-versus-host disease). Giving tacrolimus and filgrastim before or after the transplant may stop this from happening. Fludarabine, bendamustine, melphalan, and rituximab are commonly given before stem cell transplants. Giving inotuzumab ozogamicin with chemotherapy may work better in treating patients with leukemia or lymphoma undergoing stem cell transplantation.