Alemtuzumab, Fludarabine Phosphate, and Total-Body Irradiation Followed by Cyclosporine and Mycophenolate Mofetil in Treating Patients Who Are Undergoing Donor Stem Cell Transplant for Hematologic Cancer

Inclusion Criteria: * The patient must be not eligible for conventional transplants and must have disease expected to be stable for at least 100 days without chemotherapy * Patients with hematologic malignancies treatable with HCT will be included: * Aggressive non-Hodgkin lymphomas (NHLs) and other histologies such as diffuse large B-cell NHL: not eligible for autologous HCT, not eligible for conventional myeloablative HCT, or after failed autologous HCT; * Low grade NHL: with \< 6 month duration of complete response (CR) between courses of conventional therapy; * Mantle cell NHL: may be treated in first CR; * Chronic lymphocytic leukemia (CLL): must have failed 2 lines of conventional therapy and must be refractory to fludarabine; this includes patients who fail to have a complete or partial response after therapy with a regimen containing fludarabine (or another nucleoside analog\] or experience disease relapse within 12 months after completing therapy with a regimen containing fludarabine \[or another nucleoside analog); * Hodgkin's disease (HD): must have received and failed frontline therapy and have failed or were not eligible for autologous transplant; * Multiple myeloma (MM): must have received prior chemotherapy or failed autografting; following a planned autologous transplant \[tandem\] is allowed; * Acute myeloid leukemia (AML): must have \< 5% marrow blasts at the time of transplant; * Acute lymphocytic leukemia (ALL): must have \< 5% marrow blasts at the time of transplant; * Chronic myelogenous leukemia (CML): patients will be accepted beyond first clinical progression (CP1) if they have received previous myelosuppressive chemotherapy or HCT, and have \< 5% marrow blasts at time of transplant; * Myelodysplastic syndrome/myeloproliferative disease (MDS/MPD): must have failed previous myelosuppressive chemotherapy or HCT, and have \< 5% marrow blasts at time of transplant; * Waldenstrom's macroglobulinemia: must have failed 2 courses of therapy * Patient refuses to be treated on a conventional transplant protocol; for this inclusion criteria, transplants must be approved by both the participating institution's patient review committee, such as the Patient Care Conference (PCC) at the Fred Hutchinson Cancer Research Center (FHCRC), and the FHCRC principal investigator * Patient with related or unrelated donors for whom: * There is a likelihood of disease progression while HLA typing and results of a preliminary search and the donor pool suggest that a 10/10 HLA-A, B, C, DRB1 and DQB1 matched unrelated donor will not be found; * Patient and donor must be matched for at least one DRB1 allele and one DQB1 allele; * Best available matches are HLA class I HLA-A, -B, -C allele matched donors allowing for any one or two DRB1 and/or DQB1 antigen/allele mismatch; * There is no indication for an autologous transplantation as a treatment option * DONOR: For HLA matching inclusion criteria, see patient inclusion criteria * DONOR: Only peripheral blood stem cells (PBSC) will be permitted as a HSC source on this protocol Exclusion Criteria: * Positive crossmatch between donor and recipients * Patient's life expectancy is severely limited by diseases other than malignancy * Patient has central nervous system (CNS) involvement with disease refractory to intrathecal chemotherapy * Presence of circulating leukemic blasts (in the peripheral blood) detected by standard pathology for patients with AML, ALL or CML * Patient is a fertile man or woman unwilling to use contraceptives during and for up to 12 months post treatment * Patient is a female who is pregnant or breastfeeding * Patient is human immunodeficiency virus (HIV) positive * Patients with active non-hematologic malignancies (except non-melanoma skin cancers) * Patients with a history of non-hematologic malignancies (except non-melanoma skin cancers) currently in a complete remission, who are less than 5 years from the time of complete remission, and have a \> 20% risk of disease recurrence * Patient has a fungal infection with radiological progression after receipt of amphotericin B or active triazole for greater than 1 month * Patient has the following organ dysfunction: * Symptomatic coronary artery disease or ejection fraction \< 35% or other cardiac failure requiring therapy; ejection fraction is required if age \> 50 years or if the patient has a history of anthracyclines or history of cardiac disease; * Diffusion capacity of the lung for carbon monoxide (DLCO) \< 35% total lung capacity (TLC) \< 35%, forced expiratory volume of the lung in one second (FEV1) \< 35% and/or receiving supplementary continuous oxygen; the FHCRC study principal investigator (PI) must approve enrollment of all patients with pulmonary nodules; * Liver function abnormalities: patient with clinical or laboratory evidence of liver disease will be evaluated for the cause of liver disease, its clinical severity in terms of liver function, bridging fibrosis, and the degree of portal hypertension; the patient will be excluded if he/she is found to have fulminant liver failure, cirrhosis of the liver with evidence of portal hypertension, alcoholic hepatitis, esophageal varices, a history of bleeding esophageal varices, hepatic encephalopathy, uncorrectable hepatic synthetic dysfunction evinced by prolongation of the prothrombin time, ascites related to portal hypertension, bacterial or fungal liver abscess, biliary obstruction, chronic viral hepatitis with total serum bilirubin \> 3mg/dL, or symptomatic biliary disease * Patient has poorly controlled hypertension and on multiple antihypertensives * Karnofsky performance score \< 70 for adult patients * Lansky play-performance score \< 70 for pediatric patients * Patient received cytotoxic agents for "cytoreduction" within three weeks (or the interval in which a cycle of standard chemotherapy would be administered in a non-transplant setting) prior to initiating the nonmyeloablative transplant conditioning; (exceptions are hydroxyurea and imatinib mesylate) * DONOR: Marrow donors * DONOR: Positive crossmatch between donor and recipient * DONOR: Donor is HIV-positive and/or has a medical condition that would result in increased risk for filgrastim (G-CSF) mobilization and harvest of PBSC * DONOR: Donor age \< 12 years