Patient should be human leukocyte antigen (HLA) typed (A, B, C, DR and DQ) during induction therapy phase or a written explanation for not undergoing HLA typing on the flow sheet Any human leukocyte antigen (HLA) type; (historic HLA typing is permitted) Human leukocyte antigen (HLA)-A1, A2, A3, B35, or B51. No suitable human leukocyte antigen (HLA)-identical sibling donor Donor and recipient match each other for at least 7/8 human leukocyte antigen (HLA)-loci (HLA-A, B, C, and D-related [DR]). Lack human leukocyte antigen (HLA)-identical related donor Patients must have one related donor who is human leukocyte antigen (HLA) mismatched in the GVHD direction at two or more HLA loci Available human leukocyte antigen (HLA)-haploidentical donor that meets the criteria Human leukocyte antigen (HLA)-A*0201 (HLA-A2.1) positivity by molecular subtyping Human leukocyte antigen (HLA)-A*0201 (HLA-A2.1) positivity by molecular subtyping (blood test or buccal swab, historical documentation acceptable) Patients must be human leukocyte antigen (HLA)-A2+ Human leukocyte antigen (HLA)-A2 positive based on flow cytometry Absence anti-human leukocyte antigen (HLA) antibodies specific for HLA class I antigens expressed by the coagulation factor III (thromboplastin, tissue factor) (F3).cytosine deaminase (CD).carboxylesterase (CE) NSCs Patients must be human leukocyte antigen (HLA)-DP4 positive Patients must be human leukocyte antigen (HLA)-A*0201 positive Leukocyte >= 3,000/ul Human leukocyte antigen (HLA)-A*0201 (HLA-A2.1) positivity by molecular subtyping Human leukocyte antigen (HLA)-A2 positive Patients must have a related donor who is either human leukocyte antigen (HLA)-identical or a one antigen mismatch at the HLA- A; B; C; and DR loci Patients must have human leukocyte antigen (HLA)-A2 phenotype Human leukocyte antigen (HLA)-A1, -A2, -A3, or -A31 positive Any human leukocyte antigen (HLA) type Patient has anti-human leukocyte antigen (HLA) antibodies specific for HLA antigens expressed by the HB1.F3.CD NSCs Patients must express human leukocyte antigen (HLA) -A1+, -A2+, or -A3+ (80% of patients) DONOR: Human leukocyte antigen (HLA)-identical related donors or DONOR: Human leukocyte antigen (HLA) identical to recipient subject Human leukocyte antigen (HLA)-A*0201 (HLA-A2.1) positivity by molecular subtyping Human leukocyte antigen (HLA)-A2 positive Expression of human leukocyte antigen (HLA)-A:0201 or HLA-A:2402. ELIGIBILITY FOR TREATMENT ON ARM 1: Patients must express human leukocyte antigen (HLA)-A*0201 Human leukocyte antigen (HLA)-A2-positive Human leukocyte antigen HLA A2 positive Expression of human leukocyte antigen (HLA)-A2 Patients must have one related donor who is human leukocyte antigen (HLA) mismatched in the GVHD direction at two or more HLA loci Has a suitable human leukocyte antigen (HLA) haploidentical donor available Human leukocyte antigen (HLA)-A2 positive by deoxyribonucleic acid (DNA) sequence analysis (by history or as part of this study); HLA testing can be done at local labs Donor is blood-related and human leukocyte antigen (HLA)-haploidentical to the recipient DONOR: Donors will be selected to minimize human leukocyte antigen (HLA) mismatch in the host-versus-graft direction Patients must be human leukocyte antigen (HLA)-A*0201 positive Available peptide-MHC pair that can be folded into a tetramer for which MCPyV TAg-specific cells can be generated and reactivity to cell lines expressing MCPyV TAg with the corresponding human leukocyte antigen (HLA) Human leukocyte antigen (HLA)-A*02+ by serology by an ASHI accredited laboratory; Leukocyte count <3000/µL, Has a potentially suitable human leukocyte antigen (HLA) haploidentical donor available Patients must be human leukocyte antigen (HLA) A2 positive by polymerase chain reaction (PCR) typing Human leukocyte antigen (HLA) type A0201 or A2402 Patients must be human leukocyte antigen (HLA)-A*02:01-positive Human leukocyte antigen (HLA)-A2 positive. Any human leukocyte antigen (HLA) type (historic HLA typing is permitted) Patients must be human leukocyte antigen (HLA)-A*0201 positive Human leukocyte antigen (HLA)-A0201 or HLA-A2402 Availability of a willing and suitable human leukocyte antigen (HLA) identical related donor Human leukocyte antigen (HLA)-identical sibling donor Participants must be human leukocyte antigen (HLA)-A2 positive Presence of human leukocyte antigen (HLA) antibodies Availability of human leukocyte antigen (HLA)-identical sibling donor