Fludarabine Phosphate, Cyclophosphamide, Total-Body Irradiation, and Donor Bone Marrow Transplant Followed by Donor Natural Killer Cell Therapy, Mycophenolate Mofetil, and Tacrolimus in Treating Patients With Hematologic Cancer

Inclusion Criteria: * Patients with the following hematologic malignancies will be permitted although other diagnoses can be considered if approved by Patient Care Conference (PCC) and the principal investigators: * Aggressive non-Hodgkin lymphomas (NHL) and other histologies such as diffuse large B cell (DLBC) NHL - a) not eligible for autologous HCT, b) not eligible for high-dose HCT, c) after failed autologous HCT, or d) be part of a tandem auto-allo approach for high risk patients * Mantle cell NHL must be beyond first complete response (CR) * Low-grade NHL with \< 6 month duration of CR between courses of conventional therapy * Chronic lymphocytic leukemia (CLL) must have either * 1\) Failed to meet National Cancer Institute (NCI) Working Group criteria for complete or partial response after therapy with a regimen containing FLU (fludarabine phosphate) (or another nucleoside analog, e.g. 2-chlorodeoxyadenosine \[2-CDA\], pentostatin) or experience disease relapse within 12 months after completing therapy with a regimen containing FLU (or another nucleoside analog) * 2\) Failed FLU- CY (cyclophosphamide)-rituximab (FCR) combination chemotherapy at any time point; or * 3\) Have "17p deletion" cytogenetic abnormality and relapsed at any time point after any initial chemotherapy * Hodgkin lymphoma - must have received and a) failed frontline therapy, b) not be eligible for autologous HCT, or c) or be part of a tandem auto-allo approach for high risk patients * Multiple myeloma or plasma cell leukemia must have received more than one line of prior chemotherapy; consolidation of chemotherapy by autografting prior to nonmyeloablative HCT is permitted * Acute myeloid leukemia (AML) must have \< 5% marrow blasts at the time of HCT * Acute lymphocytic leukemia (ALL) must have \< 5% marrow blasts at the time of HCT * Chronic myeloid leukemia (CML) accepted if they are beyond chronic phase (CP)1 and if they have received previous myelosuppressive chemotherapy or HCT and have \< 5% marrow blasts at time of transplant * Myelodysplasia (MDS)/myeloproliferative syndrome (MPS) - (\> intermediate 1 \[int-1\] per International Prognostic Scoring System \[IPSS\]) after \> or = 1 prior cycle of induction chemotherapy; must have \< 5% marrow blasts at time of transplant * Waldenstrom's macroglobulinemia must have failed 2 courses of therapy * Patients must be expected to have disease controlled for at least 60 days after HCT * Patients for whom HLA-matched unrelated donor search could not be initiated or completed due to insurance reasons, concerns of rapidly progressive disease, and/or discretion of attending physician are eligible for this protocol * DONOR: Related, HLA-haploidentical donors who are identical for one HLA haplotype and mismatched for any number of HLA-A, -B, -C, DRB1 or DQB1 loci of the unshared haplotype * DONOR: Marrow will be the only allowed hematopoietic stem cell source * DONOR: Haploidentical donor selection will be based on standard institutional criteria, otherwise no specific prioritization will be made amongst the suitable available donors; donors will not be selected based on killer cell immunoglobulin-like receptor (KIR) status Exclusion Criteria: * Patients with available HLA-matched related donors * Patients eligible for a curative autologous HCT * Significant organ dysfunction that would prevent compliance with conditioning, GVHD prophylaxis, or would severely limit the probability of survival: * 1\) Symptomatic coronary artery disease or ejection fraction \< 35% or other cardiac failure requiring therapy (or, if unable to obtain ejection fraction, shortening fraction of \< 26%); if shortening fraction is \< 26% a cardiology consult is required with the principal investigator (PI) having final approval of eligibility * 2\) Diffusion capacity of the lung for carbon monoxide (DLCO) \< 40% total lung capacity (TLC) \< 40%, forced expiratory volume in one second (FEV1) \< 40% and/or receiving supplementary continuous oxygen; the Fred Hutchinson Cancer Research Center (FHCRC) study PI must approve enrollment of all patients with pulmonary nodules * 3\) 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 \> 3 mg/dL, or symptomatic biliary disease * Human immunodeficiency virus (HIV) seropositive patients * Patients with poorly controlled hypertension despite multiple antihypertensive medications * Fertile females who are unwilling to use contraceptive techniques during and for the twelve months following treatment, as well as females who are pregnant or actively breast feeding * Fertile males who are unwilling to use contraceptive techniques during and for the twelve months following treatment * Patients with active non-hematologic malignancies (except non-melanoma skin cancers) or those with non-hematologic malignancies (except non-melanoma skin cancers) who have been rendered with no evidence of disease, but have a greater than 20% chance of having disease recurrence within five years; this exclusion does not apply to patients with non-hematologic malignancies that do not require therapy * Active infectious disease concerns * Karnofsky performance score \< 60 Lansky performance score \< 60 * Life expectancy severely limited by diseases other than malignancy * Patients with a diagnosis of chronic myelomonocytic leukemia (CMML) * Central nervous system (CNS) involvement with disease refractory to intrathecal chemotherapy * Patients with AML, MDS, ALL, or CML must not have presence of circulating leukemic blasts detected by standard pathology * Patients with aggressive lymphomas (such as DLBC) must not have bulky, rapidly progressive disease immediately prior to HCT * Patients who have received a prior allogeneic HCT must have no active GVHD requiring immunosuppressive therapy for at least 21 days prior to start of conditioning * DONOR: Children less than 12 years of age. * DONOR: Children greater than or equal to 12 years of age who have not provided informed assent in the presence of a parent and an attending physician who is not a member of the recipient's care team * DONOR: Children greater than or equal to 12 years of age who have inadequate peripheral vein access to safely undergo apheresis * DONOR: Donors unable or unwilling to undergo marrow harvest for the initial HCT, storage of autologous blood prior to marrow harvest or apheresis one week after marrow harvest * DONOR: Donors who are not expected to meet the minimum target dose of marrow cells (1 x 10\^8 nucleated cells/kg recipient ideal body weight \[IBW\]) for the initial HCT; the average nucleated cell content of harvested marrow is 22 x 10\^6 nucleated cells/mL or 220 x 10\^8 nucleated cells/Liter * DONOR: HIV-positive donors * DONOR: Donors who are cross-match positive with recipient