Using cord blood transplant and chemotherapy to treat high-risk blood cancers

Inclusion Criteria:

* Patients aged 6 months to =\< 65 years at time of consent.
* Acute myelogenous leukemia (AML):

  * Complete first remission (CR1), complete second remission (CR2) or greater (CR2+), must have \< 5% marrow blasts at the time of transplant.
  * Patients in morphologic remission with persistent cytogenetic, flow cytometric, or molecular aberrations are eligible.
* Acute lymphoblastic leukemia (ALL):

  * Complete first remission (CR1) at high risk for relapse such as any of the following:

    * Presence of any high-risk cytogenetic abnormalities such as t(9;22), t(1;19), t(4;11) or other MLL rearrangements (11q23) or other high-risk molecular abnormality.
    * Failure to achieve MRD- complete remission after induction therapy.
    * Persistence or recurrence of minimal residual disease on therapy.
    * Any patient unable to tolerate consolidation and/or maintenance chemotherapy as would have been deemed appropriate by the treating physician.
    * Other high-risk features not defined above.
  * Complete second remission (CR2) or greater (CR2+).

    * Note: ALL with less than 5% blasts at time of transplant but persistent cytogenetic, flow cytometric or molecular aberrations are eligible.
* Other acute leukemias: Acute leukemias of ambiguous lineage or mixed phenotype with less than 5% blasts. Leukemias in morphologic remission with persistent cytogenetic, flow cytometric or molecular aberrations are eligible.
* Chronic Myeloid Leukemia (CML): Excluding refractory blast crisis. To be eligible in first chronic phase (CP1) patient must have failed or be intolerant to tyrosine kinase inhibitor therapy.
* Myelodysplastic syndromes (MDS) and myeloproliferative disorders (MPD) other than myelofibrosis:

  * MDS/MPD overlap syndromes without myelofibrosis.
  * MDS/ MPD patients must have less than 10% bone marrow myeloblasts and absolute neutrophil count (ANC) \> 0.2 (growth factor supported if necessary) at transplant work-up.
* Non-Hodgkin lymphoma (NHL) at high-risk of relapse or progression if not in remission:

  * Eligible patients with aggressive histology (such as, but not limited to, diffuse large B-cell NHL, mantle cell NHL, and T-cell histology) in CR by PET/CT imaging.
  * Eligible patients with indolent B-cell NHL (such as, but not limited to, follicular, small cell or marginal zone NHL) will have 2nd or subsequent progression with PR or CR by PET/CT imaging.
* Blastic plasmacytoid dendritic cell neoplasm (BPDCN) in morphologic remission.
* Only for adult patients, to prevent graft rejection, patients who received only non-lymphodepleting agents for their malignancy (hypomethylating agents, venetoclax, hydroxyurea, TKIs etc.), or patients who received lymphodepleting chemotherapy \> 3 months prior to scheduled admission, may receive fludarabine 25 mg/m\^2 daily x 3 days for lymphodepletion 14-42 days (aiming for 2-4 weeks) at the discretion of the principal investigator (PI).
* For patients \> 16 years old, Karnofsky score ≥ 70%. For patients =\< 16 years old, Lansky score ≥ 50%.
* Calculated creatinine clearance \> 70 ml/min.
* Bilirubin \< 1.5 mg/dL (unless benign congenital hyperbilirubinemia or hemolysis).
* Alanine transaminase (ALT) \< 3 x upper limit of normal (ULN).
* For patients \> 18 years old, pulmonary function (spirometry and corrected diffusing capacity for carbon monoxide \[DLCO\]) \> 60% predicted. For patients =\< 18 years old, or any patient unable to perform pulmonary function tests, O2 saturation \> 92% on room air.
* Left ventricular ejection fraction \> 50%.
* Albumin \> 3.0 g/dL.
* For patients \> 18 years old, Hematopoietic Cell Transplantation Comorbidity index (HCT-CI) =\< 5.
* UCB units will be selected according to current umbilical cord blood graft selection algorithm. One or two UCB units may be used to achieve the required cell dose.
* The UCB graft is matched at 4-6 HLA-A, B, DRB1 antigens with the recipient. This may include 0-2 antigen mismatches at the A or B or DRB1 loci. Unit selection based on cryopreserved nucleated cell dose and HLA-A, B, DRB1 using intermediate resolution A, B antigen and DRB1 allele typing.

Exclusion Criteria:

* Diagnosis of myelofibrosis or other malignancy with moderate-severe bone marrow fibrosis.
* Patients persistent with central nervous system (CNS) involvement in cerebrospinal fluid (CSF) or CNS imaging at time of screening0
* Prior checkpoint inhibitors/ blockade in the last 12 months.
* Two prior stem cell transplants of any kind.
* One prior autologous stem cell transplant within the preceding 12 months.
* Prior allogeneic transplantation.
* Prior involved field radiation therapy that would preclude safe delivery of 400cGy total body irradiation (TBI) in the opinion of radiation oncology.
* Active and uncontrolled infection at time of transplantation.
* HIV infection.
* Inadequate performance status/ organ function.
* Pregnancy or breast feeding.
* Patient or guardian unable to give informed consent or unable to comply with the treatment protocol including appropriate supportive care, long-term follow-up, and research tests.