Dasatinib for Modulating Immune System After Autologous Stem Cell Transplants for Multiple Myeloma, Non-Hodgkin, or Hodgkin Lymphoma

Inclusion Criteria: * Recipients of first ASCT for the treatment of hematologic malignancies (multiple myeloma, Hodgkin's and non Hodgkin's lymphoma) * Patients must be between 100 to 180 days after ASCT * Dasatinib use prior to ASCT is allowed * Performance status \>= 60% * Presence of 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 times the institutional ULN * Hemoglobin \>= 8 g/dL * Absolute neutrophil counts \>= 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: * Patients who have evidence of disease progression before day 100 after ASCT * Sex and reproductive status: * 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 * Medical history and concurrent diseases: * 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 corrected QT (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 intravenous (IV) bisphosphonates will be withheld for the first 8 weeks of dasatinib therapy due to risk of hypocalcemia * Other exclusion criteria: * Prisoners or subjects who are involuntarily incarcerated * Subjects who are compulsorily detained for treatment of either a psychiatric or physical (eg, infectious disease) illness