Donor Peripheral Blood Stem Cell Transplant in Treating Patients With Hematologic Malignancies

Inclusion Criteria: * Molecular based human leukocyte antigen (HLA) typing will be performed for the HLA-A, -B, -Cw, -DRB1 and -DQB1 loci to the resolution adequate to establish haplo-identity; a minimum match of 5/10 is required; an unrelated donor search is not required for a patient to be eligible for this protocol if the clinical situation dictates an urgent transplant; clinical urgency is defined as 6-8 weeks from referral or low-likelihood of finding a matched, unrelated donor * Acute leukemias (includes T lymphoblastic lymphoma); remission is defined as \< 5% blasts with no morphological characteristics of acute leukemia (e.g., Auer rods) in a bone marrow with \> 20% cellularity, peripheral blood counts showing ANC \> 1000/ul, including patients in complete remission with incomplete platelet recovery (CRp); if the marrow has \< 20% cellularity due to treatment related cytotoxicity, but still has \< 5% blasts, an exception may be made to include this patient up to principal investigator (PI) discretion * Acute lymphoblastic leukemia in high risk first complete remission (CR1) as defined by at least one of the following: * Adverse cytogenetics including but not limited to t(9;22), t(1;19), t(4;11), mixed lineage leukemia (MLL) rearrangements * White blood cell counts \> 30,000/mcL * Patients over 30 years of age * Time to complete remission \> 4 weeks * Presence of extramedullary disease * Minimal residual disease * Other risk factors determined by the patient's attending physician to be high risk features requiring transplantation * Acute myelogenous leukemia in high risk CR1 as defined by at least one of the following: * Greater than 1 cycle of induction therapy required to achieve remission * Preceding myelodysplastic syndrome (MDS) * Presence of fms-like tyrosine kinase receptor-3 (Flt3) abnormalities * French-American-British (FAB) M6 or M7 leukemia, or * Adverse cytogenetics for overall survival such as: * Those associated with MDS * Complex karyotype (\>= 3 abnormalities); or * Any of the following: inv(3) or t(3;3), t(6;9), t(6;11), + 8 \[alone or with other abnormalities except for t(8;21), t(9;11), inv(16) or t(16;16)\], t(11;19)(q23;p13.1)\] * Other risk factors determined by the patient's attending physician to be high risk features requiring transplantation * Acute leukemias in second (2nd) or subsequent remission * Biphenotypic/undifferentiated leukemias in first (1st) or subsequent complete remission (CR) * High-risk MDS status-post cytotoxic chemotherapy * Burkitt's lymphoma: second or subsequent CR * Chemotherapy-sensitive (at least stable disease) large cell, mantle cell or Hodgkin's lymphomas that have failed at least 1 prior regimen of multi-agent chemotherapy and are ineligible for an autologous transplant * Marginal zone B-cell lymphoma or follicular lymphoma that has progressed after at least two prior therapies (excluding single agent Rituxan) * Multiple myeloma (MM) stage II or III patients who have progressed after an initial response to chemotherapy or autologous hematopoietic stem cell transplantation (HSCT) or MM patients with refractory disease who may benefit from tandem autologous-nonmyeloablative allogeneic transplant * Left ventricular ejection fraction at rest must be \>= 35% * Bilirubin =\< 2.5 mg/dL * Alanine aminotransferase (ALT) \< 5 x upper limit of normal (ULN) * Aspartate aminotransferase (AST) \< 5 x ULN * Alkaline phosphatase \< 5 x ULN * Serum creatinine within normal range for age, or if serum creatinine outside normal range for age, then renal function (creatinine clearance or glomerular filtration rate \[GFR\]) \> 40 mL/min/1.73m\^2 * Forced expiratory volume in one second (FEV1), forced vital capacity (FVC), diffusion capacity of carbon monoxide (DLCO) (diffusion capacity) \>= 40% predicted (corrected for hemoglobin); if unable to perform pulmonary function tests, then oxygen (O2) saturation \> 92% on room air * Karnofsky/Lansky score \>= 60% * Patients who have received a prior allogeneic HSCT and who have either rejected their grafts or who have become tolerant of their grafts with no active GVHD requiring immunosuppressive therapy * Patients will undergo standard pre-transplant work-up as dictated by standard practice guidelines the results of which may be used for screening for this study * DONOR: Donors must be HLA-haploidentical first-degree relatives of the patient; eligible donors include biological parents, siblings, or children, or half-siblings * DONOR: Age \>= 12 years * DONOR: Weight \>= 40 kg * DONOR: Ability of donors \< 18 years of age to undergo apheresis without use of a vascular access device; vein check must be performed and verified by an apheresis nurse prior to arrival at the Seattle Cancer Care Alliance (SCCA) * DONOR: Donors must meet the selection criteria as defined by the Foundation for the Accreditation of Cell Therapy (FACT) and will be screened per the American Association of Blood Banks (AABB) guidelines Exclusion Criteria: * HLA-matched or single allele-mismatched donor able to donate * Pregnancy or breast-feeding * Current uncontrolled bacterial, viral or fungal infection (currently taking medication with evidence of progression of clinical symptoms or radiologic findings) * Patients with primary idiopathic myelofibrosis * DONOR: Positive anti-donor HLA antibody