New transplant method aims to reduce dangerous immune reactions

Copied word for word from the study's registry entry, so the wording is the study team's rather than ours.

Inclusion Criteria: * The patient must have a diagnosis of one of the following (one must be yes): * Acute myeloid leukemia (AML) * Acute lymphoblastic leukemia (ALL) * Chronic lymphoblastic leukemia (CLL) * Chronic myelogenous leukemia (CML) (chronic phase intolerant or unresponsive to tyrosine kinase inhibitors, accelerated phase, history of blast crisis) * Myelodysplastic syndrome (MDS) * Non-Hodgkin lymphoma (NHL) * Hodgkin lymphoma (HL) (received and failed frontline therapy or failed autologous transplantation or inability to collect enough peripheral blood stem cells \[PBSC\] for autologous hematopoietic cell transplant \[auto-HCT\]) * Multiple myeloma (MM) * Severe aplastic anemia * Histocompatible donor identified: * Related donor matched 5/6 or better (A, B, DRB1) * Unrelated donor matched 7/8 or better (A, B, C and DRB1) * Patients with severe aplastic anemia do not have disease requirements; however, if the patient has a mismatched donor, the patient must have had prior therapy with ATG. The following are eligible for study inclusion: * Patients with MDS/MPN only require \<5% myeloblast on bone marrow evaluation. * Patients with AML, ALL or CLL may be in CRi, patients with MM may be in VGPR * Patients with NHL/HL must be in CR * Have a Karnofsky performance status score of \> 50% * Diffusing capacity of the lung for carbon monoxide (DLCO) \> 40% predicted, corrected for hemoglobin and/or alveolar ventilation * Left ventricular ejection fraction \> 40% * Bilirubin =\< 3 x upper limit of normal * Liver alkaline phosphatase =\< 3 x upper limit of normal * Serum glutamic-oxaloacetic transaminase (SGOT) or serum glutamate pyruvate transaminase (SGPT) =\< 3 x upper limit of normal * Calculated creatinine clearance \> 40 cc/min by the modified Cockroft-Gault formula * Patient must be cleared pre-transplant by Radiation Oncology to be able to receive 400 cGy * Participants of child-bearing potential must agree to use adequate contraceptive methods (e.g., hormonal or barrier method of birth control; abstinence) prior to study entry; should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately * Patients who have failed a prior autologous or allogeneic transplant are eligible; however, at least 6 months must have elapsed between the start of this reduced intensity conditioning regimen and the last transplant if patient had a prior autologous or myeloablative allogeneic bone marrow transplant (BMT) * At least 2 weeks since prior radiation treatment and/or surgery. Appropriate washout of prior chemotherapy per BMT standard of care. If medication is not on the list, go by physician discretion * Participant must understand the investigational nature of this study and sign an Independent Ethics Committee/Institutional Review Board approved written informed consent form prior to receiving any study related procedure Exclusion Criteria: * Moderate to severe myelofibrosis within 60 days prior to transplant * Presence of human leukocyte antigen (HLA) antibodies to the donor within 60 days prior to transplant * Patients who in the opinion of the treating physician are unlikely to comply with the restrictions of allogeneic stem cell transplantation based on formal psychosocial screening. (i.e., serious, uncontrolled psychiatric illness/social situations that would limit compliance with study requirements) * Uncontrolled diabetes mellitus, cardiovascular disease, active serious infection or other condition which, in the opinion of treating physician, would make this protocol unreasonably hazardous for the patient * Known human immunodeficiency virus (HIV) positive * Pregnant or nursing female participants * Unwilling or unable to follow protocol requirements * Any condition which in the Investigator's opinion deems the participant an unsuitable candidate to receive study intervention