This phase II trial studies how well a 2-step approach to stem cell transplant works in treating patients with blood cancers. Giving chemotherapy and total body irradiation before a lymphocyte (white blood cell) and stem cell transplant helps stop the growth of cells in the bone marrow including normal blood-forming cells (stem cells) and cancer cells. By giving the donor cells in two steps, the dose of lymphocytes given can be tightly controlled and they can be made more tolerant to the body. When the healthy lymphocytes and 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 cells from a donor can make an immune response against the body's normal cells called graft versus host disease. Giving tacrolimus and mycophenolate mofetil may stop this from happening.

PRIMARY OBJECTIVES:

I. To assess whether providing a patient with "5+5" dosing in a 2 step matched related donor hematopoietic stem cell transplantation (HSCT) increases the percentage of patients who achieve full donor chimerism earlier as defined by 98% or greater donor T cell chimerism at 28 days post HSCT (d+28).

SECONDARY OBJECTIVES:

I. To assess day (d) +90 chimerism in patients receiving "5+5" dosing. II. To assess post HSCT relapse rates in patients receiving "5+5" dosing. III. To assess rates of grade II-IV graft versus host disease (GVHD) in patients receiving "5+5" dosing.

IV. To assess treatment-related mortality (TRM) in patients receiving "5+5" dosing.

EXPLORATORY OBJECTIVES:

I. To assess whether patients receiving "5+5" dosing have lower rates of cytomegalovirus (CMV) reactivation as compared to patients treated on Thomas Jefferson University (TJU) 08D.85 (1st Generation 2-Step Matched Related Trial).

OUTLINE:

CONDITIONING REGIMEN: Patients undergo total body irradiation (TBI) twice daily (BID) on days -9 to -6.

TRANSPLANT: Patients receive donor lymphocytes intravenously (IV) on day -6 after the last dose of TBI.

CONDITIONING REGIMEN: Patients receive cyclophosphamide IV on days -3 and -2.

TRANSPLANT: Patients undergo hematopoietic stem cell transplantation on day 0.

GVHD PROPHYLAXIS: Patients receive tacrolimus IV beginning on day -1 with taper beginning on day 42 in the absence of GVHD, a suspicion of GVHD, or previous history of GVHD requiring a taper delay. Patients also receive mycophenolate mofetil IV BID beginning on day -1 through day 28 in the absence of GVHD.

After completion of study treatment, patients are followed up at days 28, 90, 180, 270, and 365.

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.