This phase I/II trial studies the side effects and the best dose of inotuzumab ozogamicin when given together with fludarabine phosphate, bendamustine hydrochloride, and rituximab before donor stem cell transplant in treating patients with lymphoid malignancies. Giving chemotherapy drugs, such as fludarabine phosphate and bendamustine hydrochloride, before a donor peripheral blood stem cell transplant helps stop the growth of cancer cells or abnormal cell and helps stop the patient's immune system from rejecting the donor's stem cells. Immunotherapy with monoclonal antibodies, such as inotuzumab ozogamicin and rituximab, may help the body's immune system attack the cancer, and may interfere with the ability of tumor cells to grow and spread. 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. Sometimes the transplanted cell from a donor can make an immune system response against the body's normal cells. Giving fludarabine phosphate and bendamustine hydrochloride before the transplant together with anti-thymocyte globulin and tacrolimus may stop this from happening.