Inclusion Criteria:

• Only patients having received a preceding nonmyeloablative allogeneic transplantation with fludarabine/2 Gy total-body irradiation (TBI) - 4 Gy TBI or 2 Gy TBI - 4 Gy TBI conditioning from either a related or unrelated donor are eligible for this protocol
• Patients with persistent, relapsed or progressing malignancy after nonmyeloablative allogeneic transplantation (persistent disease will be defined as a failure to achieve a response as compared to baseline)
• Patients with rapidly progressing malignancies (acute myeloid leukemia [AML], acute lymphocytic leukemia [ALL], blastic phase chronic myelogenous leukemia [CML-BC] intermediate-high-grade non-Hodgkin lymphoma [NHL], Hodgkin's lymphoma or aggressive multiple myeloma [MM]) should receive salvage chemotherapy or radiation before DLI according to the recommendation made in this protocol; any form of salvage chemotherapy should be discontinued no less than 3 weeks before DLI; therapy with Gleevec or interferon (IFN)-alpha should be discontinued prior to DLI; after salvage chemotherapy restaging is performed, patients with progressive disease and patients not meeting the inclusion criteria of the study after chemotherapy will be excluded from the study; patients are allowed to receive further doses of chemotherapy after DLI administration if they are scheduled for further DLI; after additional therapy the patients must be restaged and must again meet inclusion criteria to receive further DLI
• Patients must be able to tolerate a taper of systemic steroids to a dosage of less than or equal to 0.25 mg/kg/day; all other immunosuppressive therapy must have been discontinued for at least two weeks without significant flares in GVHD (i.e., increase of acute GVHD by one or more grades)
• Patients must have persistent donor cluster of differentiation (CD)3 cells (> 5% donor CD3 cells by a deoxyribonucleic acid [DNA]-based assay that compares the profile of amplified fragment length polymorphisms [ampFLP] [or fluorescent in situ hybridization (FISH) studies or variable number tandem repeat (VNTR)])
• DONOR: Alternatively to a fresh unmodified leukapheresis product, previously collected cryopreserved peripheral blood stem cells (PBSC) after mobilization with granulocyte colony-stimulating factor (G-CSF) or cryopreserved unmodified leukapheresis product from the original donor can be used; if cryopreserved product is not available, the DLI product must be from the original donor of hematopoietic cell transplantation
• DONOR: Original donor of hematopoietic cell transplantation
• DONOR: Donor must give consent to leukapheresis
• DONOR: Donor must have adequate veins for leukapheresis or agree to placement of central venous catheter (femoral or subclavian)
• DONOR: Donor must be medically fit to undergo the apheresis procedure (institutional guidelines for apheresis)

Exclusion Criteria:

• Current grade II to IV acute GVHD or extensive chronic GVHD
• Karnofsky score < 50%
• Lansky Play-Performance Score < 40 for pediatric patients
• DONOR: Donors who are not suitable for medical reasons to donate peripheral blood mononuclear cells (PBMC) by continuous centrifugation according to the criteria of the American Association of Blood Banks (AABB)
• DONOR: Pregnancy
• DONOR: Human immunodeficiency virus (HIV) or human T-lymphotrophic virus (HTLV) infection
• DONOR: Recent immunization may require a delay