2-Step Approach to Stem Cell Transplant in Treating Participants With Hematological Malignancies

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 yea…