Biological Therapy in Treating Patients With Advanced Myelodysplastic Syndrome, Acute or Chronic Myeloid Leukemia, or Acute Lymphoblastic Leukemia Who Are Undergoing Stem Cell Transplantation

Inclusion Criteria: * Eligibility for Enrollment: * a. * i) Pre-transplant: Patients undergoing allogeneic hematopoietic stem cell transplantation for refractory anemia with excess blasts (RAEB), refractory anemia with excess blasts in transformation (RAEB-t), CML beyond chronic phase, AML beyond first remission, Philadelphia chromosome (BCR-ABL)-positive ALL at any stage, any ALL beyond first remission, primary refractory AML or ALL, therapy-related AML at any stage, or acute leukemia at any stage arising in a patient with an antecedent diagnosis of a myelodysplastic or myeloproliferative syndrome (including chronic myelomonocytic leukemia, CML, polycythemia vera, essential thrombocytosis, and agnogenic myeloid metaplasia with myelofibrosis); * ii) Post-transplant: Patients who have relapsed after transplant (morphologic, flow cytometric, cytogenetic and molecular relapse) can be offered enrollment on the protocol and may undergo therapy if it is considered possible to control their disease while waiting for the generation of study therapy * b. Patients and donors must both express an human leukocyte antigen (HLA)-allele for which it is possible to generate WT1-specific clones for * c. Patients must be able to provide blood and bone marrow samples required for this protocol * Eligibility for Prophylactic Treatment with CD8+ CTL After Transplant (Highest Risk Subgroup): At time of planned treatment, CD8+ CTL specific for WT1 must have been generated and have completed Quality Control (QC) testing * a. Patients must have had \> 5% morphologic blasts detectable in bone marrow or peripheral blood just prior to or at the time of transplant * b. Patients must have evidence of post transplant recovery of normal hematopoiesis (absolute neutrophil count \[ANC\] \> 500/mm\^3) for at least 7 days prior to the initiation of CTL infusions * c. Patients on immunosuppressive therapy for graft-versus-host disease (GVHD) are eligible for treatment if not receiving corticosteroids or if the dose of corticosteroids can be tapered to =\< the equivalent of 0.5 mg/kg/day of prednisone; the patient's symptoms have to remain stable and unlikely to increase to stage III or IV acute GVHD or chronic GVHD is unlikely to progress following the change in immunosuppressive therapy, after an appropriate monitoring period, as deemed by the patients treating physician and the principal investigator * Eligibility for Treatment with CD8+ CTL at the Time of Relapse after Transplant (All Others): At time of planned treatment, CD8+ CTL specific for WT1 must have been generated and have completed Quality Control (QC) testing * a. Patients must have evidence of recurrent disease post transplant; this includes patients with the following: * i) Morphologic relapse defined as one or more of the following: Abnormal peripheral blasts in absence of growth factor therapy; abnormal bone marrow blasts \> 5% of nucleated cells; extramedullary chloroma or granulocytic sarcoma * ii) Flow cytometric relapse defined as: the appearance in the peripheral blood or bone marrow of cells with an abnormal; immunophenotype detected by flow cytometry that is consistent with leukemia recurrence * iii) Cytogenetic relapse defined as: the appearance in one or more metaphases from bone marrow or peripheral blood cells of either a non-constitutional cytogenetic abnormality identified in at least one cytogenetic study performed prior to transplant or a new abnormality known to be associated with leukemia; (for CML) an increase in the number of Ph+ metaphases from bone marrow or peripheral blood between two consecutive samples after engraftment, or; an increase in the percentage of BCR/ABL+ cells by fluorescence in situ hybridization (FISH) between two consecutive samples after engraftment * iv) Molecular relapse defined as: one or more positive polymerase chain reaction (PCR) assays for the presence of clonotypic immunoglobulin heavy chain (IgH) or T cell receptor (TCR) gene rearrangement in patients transplanted for B-or T-cell acute lymphoblastic leukemia, respectively; one or more positive post transplant reverse transcription (RT)-PCR assays for the presence of BCR-ABL messenger ribonucleic acid (mRNA) fusion transcripts in patients transplanted for Philadelphia chromosome (BCRABL)-positive acute lymphoblastic leukemia; (for CML) a PCR assay of bone marrow (BM) or peripheral blood mononuclear cell (PBMC) positive for the presence of the BCR/ABL mRNA fusion transcript that quantitatively increases by greater than one order of magnitude on a subsequent sample * b. Patients on immunosuppressive therapy for GVHD at the time of relapse are eligible for treatment if not receiving corticosteroids or if the dose of corticosteroids can be tapered to \< the equivalent of 0.5 mg/kg/day of prednisone; the patient's symptoms have to remain stable and unlikely to increase to stage III or IV acute GVHD or chronic GVHD is unlikely to progress following the change in immunosuppressive therapy, after an appropriate monitoring period, as deemed by the patients treating physician and the principal investigator * DONOR: Both the patient and donor must have an HLA-allele which it is possible to generate WT1-specific clones for * DONOR: If a separate leukapheresis via peripheral intravenous access can be arranged, the stem cell donor will undergo leukapheresis to provide the required PBMC no sooner than 2 weeks before or after the stem cell mobilization and harvest * DONOR: If a separate leukapheresis is not possible, a portion of the PBMC from the donor's peripheral blood stem cell harvest may potentially be used to generate WT1-specific CTL clones; the feasibility of this option will depend upon the minimal cell dose required for transplantation and the presence of an excess harvest yield and the possibility of generating CTL from this product * DONOR: Some donors will be asked to provide both a separate leukapheresis and a portion of the peripheral blood mononuclear cells (PBMC) from the donor's peripheral blood stem cell harvest * DONOR: Leukapheresis donors must be age 18 or older Exclusion Criteria: * Patients for whom CD8+ CTL clones specific for WT1 have not been generated in time for planned infusion (these patients can potentially be treated later if CTL become available); Also we will exclude patients whose malignant cells do not over express WT-1, based on direct analysis of a bone marrow sample with \> 50% blasts or of leukemia cells isolated for expression analysis; in either case patients will be informed about the availability of other treatment protocols for which they might be eligible * Patients with Karnofsky performance status or Lansky play score =\< 30% * Patients with current stage III or IV GVHD unresponsive to therapy or requiring therapy with anti-CD3 mAb, prednisone \> 0.5 mg/kg/day (or corticosteroid equivalent), or other treatments resulting in the ablation or inactivation of T cells (such as other anti-T cell monoclonal antibodies); although the concurrent use of cyclosporine, FK506, or MMF is not strictly an exclusion criteria, attempts should be made to discontinue it if possible * Patients requiring concurrent therapy with hydroxyurea or other agents that may interfere with the function or survival of infused CTL clones * Patients with a preexisting nonhematopoietic organ toxicity that is deemed by the principal investigator to place the patient at unacceptable risk for treatment on the protocol * Patients with graft rejection or failure * DONOR: Medical conditions precluding either leukapheresis or blood donation may include but are not limited to: * Inadequate age or weight (leukapheresis donors must be age 18 or older, other criteria per physician discretion) * Active infection, with or without antibiotic treatment * Recent hepatitis exposure, hepatitis A or B antigenemia, or hepatitis C antibody positivity * Pregnancy or nursing; HIV or human T-lymphotropic virus (HTLV) infection * Severe cardiovascular disease (e.g., uncontrolled hypertension, recent myocardial infarction \[MI\], or unstable angina)