Allogeneic stem cell transplant with low-dose post-transplant cyclophosphamide to prevent graft-versus-host disease.

Inclusion Criteria:

* Age 18 or older at the time of study enrollment.
* Patients with acute leukemia (acute myeloid leukemia, acute lymphoblastic leukemia, mixed phenotype acute leukemia) or chronic myeloid leukemia with no circulating blasts and less than 5% blasts in the bone marrow.

Flow cytometric, polymerase chain reaction (PCR) or next generation sequencing (NGS) detected measurable residual disease is permitted.

* Patients with myelodysplastic syndrome (MDS) or chronic myelomonocytic leukemia with no circulating blasts and less than 10% blasts in the bone marrow (exception allowed due to lack of difference in outcomes with \<5% vs 5-10% blasts in this disease).
* Patients with secondary acute myeloid leukemia progressing from pre-existing myelodysplastic syndrome, myeloproliferative disease (MPN), or MDS/MPN overlap syndrome.
* Patients with relapsed or refractory chronic lymphocytic leukemia/small lymphocytic lymphoma who are indicated for allogeneic stem cell transplantation.
* Patients with lymphoma who are indicated for allogeneic stem cell transplantation, including follicular lymphoma, Hodgkin lymphoma, diffuse large B cell lymphoma (including primary mediastinal B cell lymphoma), mantle cell lymphoma, peripheral T cell lymphomas, and Richter's transformation.
* Planned reduced intensity or non-myeloablative conditioning regimen.
* Patients must have a related or unrelated peripheral blood stem cell donor as follows:

Sibling donor must be at least haploidentical using high resolution DNA-based HLA typing.

Children or parent donor must be at least haploidentical using high resolution DNA-based HLA typing. Children donors must be at least 18 years of age at the time of evaluation.

Unrelated donors must be a 7/8 or 8/8 match at HLA-A, B, C, and DRB1 at high resolution using DNA based typing.

* Cardiac function: must demonstrate at ejection fraction of at least 40%.
* Pulmonary function: must have FEV1 of at least 50% predicted, and DLCO corrected for hemoglobin of at least 40% predicted.
* Karnofsky performance status at least 70%.
* Women of childbearing potential (WOCP), defined as not surgically sterile (hysterectomy, tubal ligation, or oophorectomy) or at least 1 year postmenopausal, must have a negative serum pregnancy test before conditioning regimen.
* Female patients (unless post-menopausal or surgically sterilized) must agree to practice two effective methods of contraception simultaneously or agree to completely abstain from heterosexual intercourse from the time of signing informed consent through 12 months post-transplant.
* Male patients (even if surgically sterilized) who are partners of women of childbearing potential must agree to practice effective barrier contraception or abstain from heterosexual intercourse from the time of signing the informed consent through 12 months post-transplant.

Exclusion Criteria:

* Prior allogeneic stem cell transplant.
* Active central nervous system (CNS) involvement by malignant cells.
* Uncontrolled bacterial, viral, or fungal infections (currently taking medication with progression or no clinical improvement).
* Seropositive for human immunodeficiency virus (HIV) with detectable viral load, hepatitis B virus (HBV) or hepatitis C virus (HCV) with detectable viral load. Hepatitis B surface antibody positive due to vaccination or natural immunity are permitted. Patients previously treated for HCV and considered to be in sustained virologic remission (SVR) are allowed.
* Myocardial infarction within 6 months prior to enrollment, New York Heart Association (NYHA) class III-IV heart failure, uncontrolled angina, severe uncontrolled ventricular arrhythmias, or electrocardiographic evidence of acute ischemia.
* Pregnant or lactating female patients (unless feeding via formula). Women of childbearing potential (WOCBP) are required to have a negative serum or urine pregnancy test prior to conditioning regimen.
* Serious medical or psychiatric illness likely to interfere with participation in the study.
* Use of investigational agents.
* Haploidentical related recipient who are positive for DSA ≥ 5000 MFI by solid phase microarray method (Luminex).
* Any patient with steroid dose more than 10 mg/day within a week of registration.
* Autoimmune disorder requiring any active immunosuppression therapy.