Can a Two-Step stem cell transplant tame Graft-Versus-Host disease?

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

Inclusion Criteria: * Provide signed and dated informed consent form. * Have a hematological malignancy or any type of dyscrasia in which allogeneic HSCT is thought to be beneficial. * Have a related donor who is no more than a 1-antigen mismatch at the human leukocyte antigen (HLA)-A; B; C; DR loci in the GVHD direction with the patient. (Patients with a syngeneic donor may be treated on this therapeutic approach, but their outcomes will not be part of the statistical aims of the study. * LVEF (left ventricular end diastolic function) of \>= 45%. * DLCO (diffusing capacity of the lung for carbon monoxide) \>= 50% of predicted corrected for hemoglobin. * FEV-1 (forced expiratory volume at 1 second \>= 50% of predicted. * Serum bilirubin =\< 1.8. * Aspartate aminotransferase (AST) or alanine aminotransferase (ALT) =\< 2.5 x upper limit of normal. * Creatinine clearance of \>= 60 mL/min. * Have a Hematopoietic Cell Transplant Comorbidity Index (HCT-CI) score =\< 5 points (patients with greater than 5 points will be allowed for trial with approval of the principal investigator \[PI\] and at least 1 co-investigator \[co-I\] not on the primary care team of the patient). This is an adjustment to account for healthy patients who meet the spirit of this protocol but have histories that result in higher than HCT-CI 5 points. An example is a patient with a solid tumor malignancy in their remote history (adds 3 points to HCT-CI total) where the treatment for the malignancy occurred years to decades before and there has been complete recovery of toxicities. * Have a Karnofsky performance score (KPS) \>= 80%. * Women of reproductive potential (defined as women under the age of 50 years still menstruating within 2 months of HSCT despite past history of chemotherapy) will be counseled to use highly effective contraception including oral, intramuscular (IM), or patch contraceptives, intrauterine device (IUD), diaphragm, cervical cap, or contraceptive implant. Pharmacological avoidance of pregnancy and suppression of menstruation may be instituted during the HSCT inpatient stay. * Men will be asked to abstain from sexual relations during the treatment period of the HSCT stay. * DONOR: All donors are selected and screened for their ability to provide adequate infection-free apheresis products for the patient in a manner that does not put the donor at risk for negative consequences. Exclusion Criteria: * Be human immunodeficiency virus (HIV) positive. * Be pregnant or breastfeeding. * Have received alemtuzumab or rabbit antithymocyte globulin (ATG) within 8 weeks or horse ATG within 6 weeks of the transplant admission. This exclusion criterion will be documented by the absence of these drugs in the medical record. * Patients with evidence of another malignancy, exclusive of a skin cancer that requires only local treatment, should not be enrolled on this protocol without the specific approval of the PI. If the PI disregards this criterion (example of this is localized prostate cancer not yet requiring treatment), the rationale must be documented in the study binder). This exclusion criterion will be documented by the absence of these drugs in the medical record.