High Dose Cyclophosphamide, Tacrolimus, and Mycophenolate Mofetil in Preventing Graft Versus Host Disease in Patients With Hematological Malignancies Undergoing Myeloablative or Reduced Intensity Donor Stem Cell Transplant

Inclusion Criteria: * Patients with acute leukemia or chronic myelogenous leukemia with no circulating blasts and with less than 10% blasts in the bone marrow * Patients with myelodysplastic syndrome (MDS) with intermediate-2 or high risk per International Prognostic Scoring System (IPSS) (or intermediate, high, very high risk by Revised International Prognostic Scoring System \[IPSS-R\]) or myeloproliferative neoplasm; primary or secondary if high-risk features or refractory disease * Patients with chronic lymphocytic leukemia/small lymphocytic lymphoma, follicular, marginal zone, diffuse large B-cell, Hodgkin lymphoma, or mantle cell lymphoma with chemosensitive disease at time of transplantation; all types of lymphoma are eligible * High risk, or refractory and relapsed multiple myeloma * No available human leukocyte antigen (HLA)-matched related donor * Available matched unrelated donor * Ejection fraction at rest \>= 50% * Karnofsky performance status (KPS) \>= 70 * Measured creatinine clearance more than 60 mL/min. The updated Schwartz formula should be used for pediatric patients (\>=5 to 12 years old) * Carbon monoxide diffusing capability test (DLCO) \>= 50% (adjusted for hemoglobin) and forced expiratory volume in 1 second (FEV1) \>= 50% * Total bilirubin \< 1.5 x the upper limit of normal; patients who have been diagnosed with Gilbert's disease are allowed to exceed the defined bilirubin value of 1.5 x the upper limit of normal * Alanine aminotransferase (ALT)/aspartate aminotransferase (AST) \< 2.5 x the upper limit of normal * Alkaline phosphatase \< 2.5 x the upper limit of normal * Female subjects (unless postmenopausal for at least 1 year before the screening visit, or surgically sterilized), agree to practice two (2) effective methods of contraception at the same time, or agree to completely abstain from heterosexual intercourse, from the time of signing of the informed consent through 12 months post-transplant * Male subjects (even if surgically sterilized), of partners of women of childbearing potential must agree to one of the following: practice effective barrier contraception, or abstain from heterosexual intercourse from the time of signing the informed consent through 12 months post-transplant * All subjects must have the ability to understand and the willingness to sign a written informed consent document DONOR INCLUSION CRITERIA * 7 out of 8 at high resolution using deoxyribonucleic acid (DNA)-based typing with either antigen or allele mismatched HLA (-A, -B, -C, and -DR) or 8/8 HLA-mismatched with either double DQ mismatch (10/12) or combined DQ and DP mismatch * Donor must be willing to donate peripheral blood stem cells * Suitable donor * Medically cleared to donate per National Marrow Donor Program (NMDP) * Absence of donor-specific antibodies (DSA) to the mismatched HLA-locus * Donor choices per matched unrelated donor (MUD) committee according to center standard operating procedure (SOP) Exclusion Criteria: * Prior allogeneic transplant * Active central nervous system (CNS) involvement by malignant cells * Patients with uncontrolled bacterial, viral or fungal infections (currently taking medication and with progression or no clinical improvement) at time of enrollment * Patients with transformed lymphoma (e.g., Richter's transformation arising in follicular lymphoma or chronic lymphocytic leukemia) * Patients seropositive for the human immunodeficiency virus (HIV) * Patients with active hepatitis B or C determined by polymerase chain reaction (PCR) * Myocardial infarction within 6 months prior to enrollment or New York Heart Association (NYHA) class III or IV heart failure, uncontrolled angina, severe uncontrolled ventricular arrhythmias, or electrocardiographic evidence of acute ischemia or active conduction system abnormalities; prior to study entry, any electrocardiography (ECG) abnormality at screening must be documented by the investigator as not medically relevant * Female patients who are lactating or pregnant * Patients with a serious medical or psychiatric illness likely to interfere with participation in this clinical study * History of another primary malignancy that has not been in remission for at least 3 years (the following are exempt from the 3-year limit: non-melanoma skin cancer, fully excised melanoma in situ \[Stage 0\], curatively treated localized prostate cancer, and cervical or breast carcinoma in situ on biopsy or a squamous intraepithelial lesion on PAP smear) * Psychosocial issues: no appropriate caregivers identified, or non-compliant to medications * Subjects, who in the opinion of the investigator, may not be able to comply with the safety monitoring requirements of the study