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 the presence of study personnel * Written consent will include a Health Insurance Portability and Accountability Act (HIPAA) form according to institutional guidelines * Patient should be able to take oral medication (dasatinib must be swallowed whole) Exclusion Criteria: * Recipient of mismatched (allele or antigen level) graft in more than one loci of HLAA, -B, -C or -DR loci will not be eligible, i.e. recipients of 2-antigen or 2-allelele mismatched graft * Patients on investigational therapy for graft-versus-host disease (GVHD) * Patients with uncontrolled acute or chronic GVHD or refractory disease not responding to conventional therapy * Patients who have evidence of disease progression before day 100 after ASCT * Women of childbearing potential (WOCBP) who are unwilling or unable to use an acceptable method to avoid pregnancy for the entire study period and for at least 4 weeks after the last dose of study drug * Women who are pregnant or breastfeeding * Women with a positive pregnancy test * Sexually active fertile men not using effective birth control if their partners are WOCBP * No malignancy \[other than the one treated in this study\] which required radiotherapy or systemic treatment within the past 5 years * Concurrent medical condition which may increase the risk of toxicity, including: * Pleural or pericardial effusion of any grade at the time of screening for study * Cardiac Symptoms; any of the following should be considered for exclusion: * Uncontrolled angina, congestive heart failure or myocardial infarction (MI) within (6 months) * Diagnosed congenital long QT syndrome * Any history of clinically significant ventricular arrhythmias (such as ventricular tachycardia, ventricular fibrillation, or Torsades de pointes) * Prolonged QTc interval on pre-entry electrocardiogram (\> 450 msec) * History of significant bleeding disorder unrelated to cancer, including: * Diagnosed congenital bleeding disorders (e.g., von Willebrand's disease) * Diagnosed acquired bleeding disorder within one year (e.g., acquired anti-factor VIII antibodies) * Ongoing or recent (=\< 3 months) significant gastrointestinal bleeding * Any previous history of \>= grade 3 toxicity to Dasatinib * Prohibited treatments and or therapies: * Category I drugs that are generally accepted to have a risk of causing Torsades de Pointes including: (Patients must discontinue drug 7 days prior to starting dasatinib) * Quinidine, procainamide, disopyramide * Amiodarone, sotalol, ibutilide, dofetilide * Erythromycin, clarithromycin * Chlorpromazine, haloperidol, mesoridazine, thioridazine, pimozide * Cisapride, bepridil, droperidol, methadone, arsenic, chloroquine, * Domperidone, halofantrine, levomethadyl, pentamidine, sparfloxacin, lidoflazine * Patient agrees to discontinue St. Johns Wort while receiving dasatinib therapy (discontinue St. Johns Wort at least 5 days before starting dasatinib) * Patient agrees that IV bisphosphonates will be withheld for the first 8 weeks of dasatinib therapy due to risk of hypocalcemia * Prisoners or subjects who are involuntarily incarcerated * Subjects who are compulsorily detained for treatment of either a psychiatric or physical (egg, infectious disease) illness