Allogeneic Hematopoietic Cell Transplantation for Disorders of T-cell Proliferation and/or Dysregulation

* INCLUSION CRITERIA - RECIPIENT: * Age \>= 4 years * T-cell proliferation and/or dysregulation (TCP/D) deemed to be of sufficient past severity to warrant hematopoietic cell transplantation (HCT) that meets at least one of the criteria below: * Identified germline T-cell activating mutation in the phosphoinositide 3-kinase (PI3k) pathway * Identified adenosine deaminase 2 (ADA2) deficiency (biallelic mutations in CECR1 (ADA2) and/or phenotypically with low ADA2 level) leading to neutropenia requiring chronic granulocyte colony-stimulating factor (GCSF) therapy or to transfusion-dependent anemia or thrombocytopenia * T-cell infiltration of liver, spleen, lymph nodes, marrow, lungs, gut, or other organs by T cells, as evidenced by laboratory, radiographic, and/or anatomic pathology evaluation, resulting in organ dysfunction and/or organomegaly * Latent herpesvirus infection in T lymphocytes * History of or active evidence of hemophagocytic lymphohistiocytosis (HLH) * Recurrent or prolonged fevers attributed to immune dysregulation * T-cell population in blood and/or marrow with immunophenotype of large granular lymphocytes (LGL), with or without clonality or lymphocytosis * T-cell lymphoproliferative disorder in the setting of an underlying immune defect * Immune-mediated cytopenias of one lineage requiring transfusion or GCSF support or of 2 or 3 lineages with or without transfusion or support * Chronic active Epstein-Barr virus (EBV) * At least one potential 7-8/8 human leukocyte antigen (HLA)-matched related (excluding an identical twin) or unrelated donor (at HLA-A, -B, -C, and -DR), or an HLA-haploidentical related donor, based on initial low resolution unrelated donor search and/or at least one biologically- related family member who has at least a 25% chance of being at minimum an HLA- haploidentical match and is potentially suitable to donate based on reported family history. HLA typing of potential donors and/or mutation testing does not need to be completed for eligibility. * Adequate end-organ function, as measured by: * Left ventricular ejection fraction (LVEF) greater than or equal to 40% by 2-dimensional (2D) echocardiogram (ECHO) or left ventricular shortening fraction greater than or equal to 20% by ECHO for subjects receiving reduced-intensity conditioning (RIC), or LVEF greater than or equal to 30% if the subject has radiologic evidence of aortic, renal, or coronary artery vasculitis. LVEF greater than or equal to 30% for subjects receiving immunosuppression-only conditioning (IOC). * Pulmonary function tests: diffusing capacity of the lungs for carbon monoxide (DLco) (corrected for hemoglobin) and forced expiratory volume (FEV1) greater than or equal to 40% of predicted for the RIC arm, and greater than or equal to 30% predicted for the IOC arm; or in pediatric subjects, if unable to perform pulmonary function tests, there should be no evidence of dyspnea at rest, no requirement for supplemental oxygen, and oxygen saturation \>92% on room air. Calculations will be based on the values reported in (Clinical Research Information System (CRIS). * Bilirubin \<= 3.0 mg/dL (unless due to Gilbert's syndrome or hemolysis) for subjects receiving RIC and bilirubin \<= 5.0 mg/dL for subjects receiving IOC (unless due to Gilbert's syndrome or hemolysis); alanine aminotransferase (ALT) and aspartate aminotransferase (AST) \<= 5 x upper limit of normal (ULN) for subjects receiving RIC or \<= 10 x ULN for subjects receiving IOC. Subjects who are above these bilirubin, ALT, or AST thresholds may be eligible for the RIC or IOC arm if evaluated by a hepatologist who deems the liver function test abnormalities to be potentially reversible with HCT. * Estimated creatinine clearance of \>= 50 mL/min/1.73 m\^2, calculated using estimated glomerular filtration rate (eGRF) in the clinical lab for adults and the Schwartz formula for pediatric subjects, if eGFR not reported by the clinical lab. * Karnofsky (adults) or Lansky (children) performance status of \>= 50% or Eastern Cooperative Oncology Group (ECOG) performance status of 2 or less for the RIC arm and \>=30% or ECOG performance status of 3 or less for the IOC arm * Ability of subject or parent/legal guardian or Legally Authorized Representative (LAR) (e.g., in cases of adults unable to consent) to understand and the willingness to sign a written informed consent document * Not pregnant or breastfeeding. As therapeutic agents used in this trial may be harmful to a fetus, women of childbearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and for at least one-year post-allo HCT. Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in the study, she should inform her treating physician immediately. * Disease status: Subjects with lymphoproliferative disorder (LPD), large granular lymphocytic leukemia (LGL), hemophagocytic lymphohistiocytosis (HLH), or other TCP/D disorders requiring standard therapies to prepare for HCT should be referred in remission if possible. However, these diseases are often aggressive and require swift evaluation for HCT while concurrently attempting to establish disease control through the administration of standard therapies. If ongoing therapy for the underlying disease outside of the National Institutes of Health (NIH) is not in the best interest of the subject according to the clinical judgment of the principal investigator (PI), then the subject may receive standard treatment for his/her underlying TCP/D disorder as a bridge to HCT on this protocol, prior to starting the research phase of the study. If it becomes apparent that the subject will not be able to proceed to HCT, then he/she must come off study. Subjects receiving standard therapy will be told about the therapy, associated risks, potential benefits, alternatives to the proposed therapy, and the availability of receiving the same treatment elsewhere, outside of a research protocol. EXCLUSION CRITERIA - RECIPIENT: * Subjects who are receiving any other investigational agents, with the exception of virus- specific cytotoxic T-cells for the treatment of viral infection/reactivation prior to allo HCT. * Prohibitive allergy to a study drug or to compounds of similar chemical or biologic composition of the agents (equine anti-thymocyte globulin (e-ATG), steroids, cyclophosphamide, busulfan, pentostatin, tacrolimus, mycophenolate mofetil (MMF), G-CSF) used in the study. * Active psychiatric disorder which is deemed by the PI to have significant risk of compromising compliance with the transplant protocol, or which does not allow for appropriate informed consent * Human immunodeficiency virus (HIV) positive or other acquired immunodeficiency that, as determined by the PI, interferes with the assessment of TCP/D severity and/or the attribution of clinical manifestations of immunodeficiency to a disorder of TCP/D. * Magnesium Transporter 1 (MagT1) mutation and active need to take anti-platelet agents and/or therapeutic anti- coagulation that cannot be interrupted during aplasia * Lack of adequate central venous access potential INCLUSION CRITERIA RELATED DONOR * Age greater than or equal to 4 years * Related donor deemed suitable and eligible, and willing to donate, per clinical evaluations who are additionally willing to donate blood, urine, and marrow specimens for research. Related donors will be evaluated in accordance with existing Standard Policies and Procedures for determination of eligibility and suitability for clinical donation. Note that participation in this study is offered to all related donors, but is not required for clinical donation, so it is possible that not all related donors will enroll onto this study. EXCLUSION CRITERIA - RELATED DONOR: -None INCLUSION CRITERIA - UNRELATED DONOR: -Unrelated donors will be evaluated in accordance with existing National Marrow Donor Program (NMDP) Standard Policies and Procedures, available at: http://bethematch.org/About-Us/Global- transplant-network/Standards/, except for the additional requirement of EBV serostatus testing for clinical purposes of donor selection. Note that participation in this study is offered to all unrelated donors but not required for clinical donation, so it is possible that not all unrelated donors will enroll on this study. Unrelated donors only enroll if they contribute research specimens, which is optional. EXCLUSION CRITERIA - UNRELATED DONOR: -Unrelated donors: failure to qualify as a National Marrow Donor Program (NMDP) donor per current NMDP Standards, available at: http://bethematch.org/About-Us/Global-transplant-network/Standards/. Exceptions to donor eligibility (e.g. foreign travel, tattoos) do not automatically exclude the donor and will be reviewed by the PI.