Dasatinib for Immune Modulation After Donor Stem Cell Transplant for Hematologic Malignancies

Inclusion Criteria: * Recipients of first ASCT from related or unrelated donor for the treatment of hematologic malignancies (acute myeloid leukemia, chronic myeloid leukemia, acute lymphoblastic leukemia, chronic lymphocytic leukemia, myelodysplastic syndrome, Hodgkin and non-Hodgkin lymphoma) who are 1-antigen or 1-allele mismatched or fully matched at human leukocyte antigen (HLA)-A, -B, -C and -DR as defined by high resolution typing * Patients must be between 100 - 180 days after allogeneic stem cell transplantation * Dasatinib use prior to ASCT is allowed * Performance status \>= 60% * Presence of large granular lymphocyte (LGL) clone prior to enrollment will not be an exclusion criterion if the LGL clone is \< 25% of T cell population * Total bilirubin \< 2.0 times the institutional upper limit of normal (ULN) * Hepatic enzymes (aspartate aminotransferase \[AST\], alanine aminotransferase \[ALT\] ) =\< 2.5 times the institutional ULN * Serum creatinine \< 1.5 time the institutional ULN * Hemoglobin \>= 8 g/dL * Absolute neutrophil count 1,500 cells per uL * Platelets \>= 100,000 per uL * Patient should be able to provide signed written informed consent: * Before any study procedures are performed, subjects will have the details of the study described to them, and they will be given a written informed consent document to read; then, if subjects consent to participate in the study, they will indicate that consent by signing and dating the informed consent document in…