Engineered Donor Stem Cell Transplant in Treating Patients With Hematologic Malignancies

Inclusion Criteria: * Lack of a human leukocyte antigen (HLA) matched related donor, lack of an immediately available 8/8 HLA matched unrelated donor * Patients must be diagnosed with a high-risk and/or advanced hematologic malignancy defined as one of the following * Acute lymphocytic leukemia (ALL) in complete remission (CR)1 with high-risk features including adverse cytogenetic such as t(9;22), t(1;19), t(4;11), or MLL gene rearrangements; in second or greater morphologic remission; persistent minimal residual disease * Acute myeloid leukemia (AML) in CR1 with intermediate-risk disease and persistent detectable minimal residual disease (MRD), or with high-risk features defined as: greater than 1 cycle of induction therapy required to achieve remission; preceding myelodysplastic syndrome (MDS) or myeloproliferative disease; presence of FLT3 mutations or internal tandem duplications, DNMT3a, TET2, MLL-partial tandem duplication (PTD), ASXL1, PHF6; FAB M6 or M7 classification; adverse cytogenetics including: -5, del 5q, -7, del7q, abnormalities involving 3q, 9q, 11q, 20q, 21q, 17, +8, complex (\> 3 abnormalities) * Patients with AML must have less than 10% bone marrow blasts and \< 100/mcL absolute peripheral blood blast count * Patients with AML in CR2, subsequent CR or with active disease at transplant (\< 10% bone marrow blasts) * MDS with International Prognostic Scoring System (IPSS) intermediate-2 or higher, therapy-related MDS or chronic myelomonocytic leukemia (CMML) * Aplastic anemia with absolute neutrophil count (ANC) \< 1,000 and transfusion dependent after failed immunosuppression therapy * Chronic myeloid leukemia (CML) \>= 1st chronic phase, after failed \>=2 lines of tyrosine kinase inhibitors; patients who progressed to blast phase must be in morphologic remission at transplant * Relapsed Hodgkin's disease or non-Hodgkin's lymphoma (NHL) * Patients with chemo-sensitive chronic lymphocytic leukemia (CLL)/small lymphocytic lymphoma (SLL) with persistent or recurrent disease after fludarabine-based regimens with \< 25% involvement by CLL/SLL cells * Patients with lymphoblastic lymphoma in remission or after partial response to chemotherapy * Patients with poor prognosis multiple myeloma by cytogenetics del13, del 17p, t(4;14) or t(14;16) or hypodiploidy, with advanced disease (stage \>= 2) and /or relapsed after autologous stem cell transplant * Zubrod performance status 0-1 or Karnofsky performance status \> 70%; patients \> 50 years will have to have a Sorror Comorbidity Index =\< 3 * Available haploidentical donor willing and eligible to undergo a peripheral blood collection * Left ventricular ejection fraction (LVEF) \> 40% * Bilirubin =\< 1.5 mg/dl (unless Gilbert's syndrome), alanine aminotransferase (ALT) or aspartate aminotransferase (AST) =\< 200 IU/ml for adults; conjugated (direct) bilirubin \< 2 x upper limit of normal * Serum creatinine clearance \>= 50 ml/min (calculated with Cockcroft-Gault formula) * Diffusing capacity for carbon monoxide (DLCO) \>= 45% predicted corrected for hemoglobin * Patient or patient's legal representative must provide written informed consent Exclusion Criteria: * Human immunodeficiency virus (HIV) positive; active hepatitis B or C * Patients with active infections; the principal investigator (PI) is the final arbiter of the eligibility * Liver cirrhosis with greater than grade 1 stage 1 inflammation/fibrosis * Uncontrolled central nervous system (CNS) involvement by tumor cells within the past 2 months * History of another primary malignancy that has not been in remission for at least 3 years; (the following are exempt from the 3-year limit: nonmelanoma skin cancer, fully excised melanoma in situ \[stage 0\], curatively treated localized prostate cancer, and cervical carcinoma in situ on biopsy or a squamous intraepithelial lesion on PAP smear) * Positive beta human chorionic gonadotropin (HCG) test in a woman with child bearing potential defined as not post-menopausal for 12 months or no previous surgical sterilization * Inability to comply with medical therapy or follow-up