Donor Stem Cell Transplant in Treating Patients With High-Risk Chronic Lymphocytic Leukemia or Small Lymphocytic Lymphoma

OBJECTIVES: Primary * To determine if this treatment can improve 2-year current progression-free survival (PFS) in the early disease cohort compared to historical controls. Specifically, we plan to study whether we can achieve 2-year PFS ≥ 70% and to exclude 2 year PFS ≤ 50% Secondary * To determine whether in the advanced disease cohort we can achieve 2-year current PFS ≥ 50% and to exclude 2-year PFS ≤ 30% * To assess objective response rate. * To assess the incidence of grade 2-4 and 3-4 acute graft-vs-host disease (GVHD). * To assess the incidence of extensive chronic GVHD. * To assess the incidence of treatment-related mortality at 100 days and 1 year * To assess overall survival * To assess donor chimerism for CD3+ cells at 1 and 2 years after transplantation * To investigate the presence of donor antigen-specific T-cell clones before and after withdrawal of immune suppression. * To compare the relapse profiles of patients with T-cell responses against CLL to those whose CLL cells are not reactive * To prospectively examine the impact of high-risk genomic features and immune-based single nucleotide polymorphisms on response, toxicity, and 2-year PFS to reduced intensity allogeneic stem cell transplant OUTLINE: This is a multicenter study. * Preparative regimen: Patients receive 1 of 2 preparative regimens at the discretion of the participating institution. * Preparative regimen 1: Patients receive rituximab IV on days -7, -1, 7, and 14 and fludarabine phosphate IV over 30 minutes and busulfan IV over 3 hours on days -5 to -2. . * Preparative regimen 2: Patients receive rituximab IV on days -7, -1, 7, and 14, fludarabine phosphate IV over 30 minutes on days -5 to -2, and cyclophosphamide IV over 1-2 hours on days -5 to -3. Patients with matched unrelated donors also receive anti-thymocyte globulin IV over 4-6 hours on days -6 to -4. * Graft-vs-host disease (GVHD) prophylaxis: Patients who receive preparative regimen 1 may receive either GVHD prophylaxis regimen 1 or 2; patients who receive preparative regimen 2 may only receive GVHD prophylaxis regimen 2. * GVHD prophylaxis regimen 1: Patients receive tacrolimus either orally or IV and oral sirolimus beginning on day -2 and continuing until day 60, followed by a taper until day 180. Patients also receive methotrexate IV on days 1, 3, and 6. * GVHD prophylaxis regimen 2: Patients receive tacrolimus either orally or IV beginning on day -2 and continuing until day 60, followed by a taper until day 180. Patients also receive methotrexate IV on days 1, 3, 6, and 11. * Transplantation: Patients undergo allogeneic peripheral blood stem cell transplantation on day 0. * Maintenance therapy: Patients receive rituximab IV at 3, 6, 9, and 12 months after transplantation. Peripheral blood and bone marrow aspirate samples may be collected periodically for correlative laboratory studies. Patients are followed up periodically for a maximum of 5 years from study entry.