Haploidentical Stem Cell Transplant with Prophylactic Natural Killer DLI for Lymphoma, Multiple Myeloma, and CLL

General Inclusion Criteria (For Treatment Groups) * Patient age 18 - 75 years * ECOG 0 - 2 * HIV-positive patients are allowed if these criteria are met: 1. No history of opportunistic infections 2. CD4+ cell count greater or equal to 250 cells/mm3 3. No history of non-malignancy AIDS-defining conditions other than historical low CD4+ cell counts 4. Patient is on antiretroviral therapy with undetectable viral load. There must be minimal interactions of the antiviral therapy with the experimental treatment (antiretroviral such as ritonavir is potent CYP3A4 inhibitor and p-gp inducer may interact with tacrolimus resulted in increased serum concentration of tacrolimus). * Patients must have a related donor or who is at minimum HLA haploidentical. The donor and recipient must be identical at least one allele of each of the following genetic loci: HLA-A, HLA-B, HLA-Cw, HLA-DRB1, and HLA-DQB1. A minimum match of 5/10 is therefore required, and will be considered sufficient evidence that the donor and recipient share one HLA haplotype. An unrelated donor search is not required. (Patients with a readily-available, suitable, fully-matched sibling donor less than age 55 are not eligible for this trial, these patients should proceed to transplant using the matched related donor as standard-of-care). Criteria for Donor Eligibility * Age greater than 12 years * Donors must meet the selection criteria as defined by the Foundation for the Accreditation of Hematopoietic Cell Therapy (FACT). * In the event that two or more eligible donors are identified, the following order of priority will be used to determine the preferred donor: * Medically and psychologically fit and willing to donate * For CMV seronegative recipients, a CMV seronegative donor * Red blood-cell compatibility * RBC cross-match compatible * Minor ABO incompatibility * Major ABO incompatibility * If more than one preferred donor is identified and there is no medical, HLA- or KIR ligand reason to prefer one of them, then the following guidelines are recommended: If the patient is male, choose a male donor: * Choose the youngest preferred donor * If the patient and family express a strong preference for a particular donor, use that one. Inclusion Criteria (Lymphoma) * Diagnosis of resistant or relapsed CLL, Non-Hodgkin Lymphoma, Hodgkin Lymphoma, T-cell lymphoma, NK or NK/T Lymphoma. * Meets one of the following criteria: * relapsed after auto-transplant, or * failed to mobilize autologous stem cells, or * for whom allogeneic stem cell transplant is deemed appropriate given disease risk factors that make cure with autologous transplant seem unlikely, such as history of chemotherapy refractoriness, high risk disease features/mutations/translocations (e.g., Double Hit / Double Expressor DLBCL), short remission after prior chemotherapy, or histologic transformation (see below). * For Patients with Aggressive Mantle Cell and Diffuse Large B Cell Lymphoma who have not had a prior autologous transplant: * Must have received at least 2 lines of prior therapy, and * Have been exposed to anthracycline, and * High and High-Intermediate aaIPI score (2 or 3 factors), and * Have relapsed within one year of primary therapy * For diagnosis of other aggressive lymphoma (e.g. NK/T Lymphoma, T Cell Lymphoma, etc.): * Must have received at least 2 lines of prior therapy, and * Relapsed within 12 months of most recent therapy * For low-grade lymphomas / CLL: * Standard risk patients (absence of del(17p), absence of del(11q), no TP53 mutation and absence of complex karyotype) must have progressed on BCR inhibitor, or undergone histologic transformation, to be eligible. * Patients with high risk disease (del(17p) or TP53 mutations and/or complex phenotype) who relapse after frontline therapy, demonstrate refractory disease to second line therapy (not BCR inhibitors), but show an objective response to BCR inhibitors are eligible to be taken off BCR inhibitors in order to proceed to alloHSCT on trial. Patients with high risk disease who relapse after frontline therapy, demonstrate refractory disease to second line therapy including BCR inhibitors (not BCL-2 inhibitors), but show an objective response to BCL-2 inhibitors (venetoclax) are eligible to be taken off BCL-2 inhibitors in order to proceed to alloHSCT on trial. * For aggressive lymphomas, partial or complete remission (PR or CR) is required prior to alloHSCT. * Regarding CD20 expression: Patients with B cell malignancies that were CD20+ at any level at the time of relapse diagnosis (including partial / dim staining on IHC or partial / low level expression by flow cytometry) will receive rituximab as part of allogeneic transplant conditioning, if indicated. Patients with primary-refractory disease who were CD20+ at any level at the time of diagnosis will likewise receive rituximab, if indicated. Patients with histologies that were CD20- will not receive rituximab (T cell lymphoma, NK/T lymphoma, etc.). Fresh tissue / repeat biopsy is not required; the most recent biopsy will be reviewed to assess CD20 status. Inclusion Criteria (Multiple Myeloma) \- Patient age 18 - 75 years with: * Early relapse (less than 24 months) after primary therapy that included an autologous HSCT, or * High risk multiple myeloma defined as t(4;14), del(17p), -13, t(14;16), amp (1q21), chromosome 8q24.1/c-MYC abnormality, or LDH \> ULN at diagnosis, provided patients respond favorably to salvage therapy before enrollment for alloHSCT on trial and patient is age \< 55, or * Patients failing to mobilize peripheral blood stem cells for autologous transplantation, or * Extramedullary disease at diagnosis or relapse, or * Plasma-cell leukemia with chemosensitive disease Inclusion Criteria - Control Patients (specimen collection, only) * Age 18-75 years * Undergoing standard-of-care reduced-intensity peripheral blood allogeneic stem cell transplantation (any indication, donor source, conditioning regimen) using PTCy GVHD prophylaxis. * Willing to provide longitudinal blood samples per Control Specimen Collection Calendar for correlative studies (for comparison to specimens from patients treated on the trial). * Agrees to let study personnel collect excess bone marrow aspirate whenever a bone marrow biopsy is performed for clinical purposes, and use for research. Exclusion Criteria * Patient has a readily-available, suitable, fully-matched sibling donor (MRD) less than age 55. 'Suitable' means no high-titre donor-specific antibodies present, and negative IDM testing with no contraindications. * Patient has a clinically-significant donor-specific antibody for the selected donor (DSA clearance is not allowed). * Poor cardiac function: left ventricular ejection fraction \<45% as determined by MUGA or ECHO. * Symptomatic pulmonary disease. Poor pulmonary function: FEV1, FVC, and DLCO \<50% predicted (corrected for hemoglobin) for patients who have not received thoracic or mantle irradiation. For patients who have received thoracic or mantle irradiation, FEV1 and FVC \<70% predicted or DLCO \< 50 of predicted. * Poor liver function: bilirubin \>2 mg/dl (not due to hemolysis, Gilbert's or primary malignancy). ALT or AST \> 5 x laboratory upper normal limits. * Poor renal function: Creatinine \>2.0mg/dl or creatinine clearance (calculated creatinine clearance is permitted) \< 60 mL/min based on Traditional Cockcroft-Gault formula * Women of childbearing potential who currently are pregnant (Β-HCG+) or who are not practicing adequate contraception. * Uncontrolled viral, bacterial, or fungal infections. Patients with symptoms consistent with RSV, influenza A, B, or parainfluenza at the time of enrollment will be assayed for the above viruses and if positive are not eligible for the trial until they are no longer symptomatic (patients may have continued assay positivity for a period of time post resolution of symptoms secondary to the nature of the assay). * Uncontrolled CNS involvement by malignancy (patients with prior history of CNS disease controlled with intrathecal chemotherapy or prior systemic therapy are allowed). * Patients who have any debilitating medical or psychiatric illness which would preclude their giving informed consent or their receiving optimal treatment and follow-up. Exclusion Criteria - Control Patients (specimen collection, only) * Undergoing myeloablative alloHSCT. * Non-PTCy GVHD prophylaxis. * Non-PBSC transplant (bone marrow stem cell source). * Not willing to give longitudinal blood specimens for research use or not willing to allow access to medical records for non-clinical purposes.