Comparing ATG or Post-Transplant Cyclophosphamide to Calcineurin Inhibitor-Methotrexate as GVHD Prophylaxis After Myeloablative Unrelated Donor Peripheral Blood Stem Cell Transplantation

Inclusion Criteria: * The following diseases will be permitted, although other diagnoses can be considered if approved by Fred Hutch Patient Care Conference or the participating institution's patient review committees and the principal investigator: * Acute lymphocytic leukemia (ALL) in complete remission (CR)1 with high risk features defined as evidence of adverse cytogenetics such as t(9;22), t(1;19), t(4;11), or MLL rearrangements or presence of minimal residual disease * Acute myeloid leukemia (AML) in CR1 with high risk features defined as: * Intermediate or adverse risk disease as defined by European LeukemiaNet (ELN) 2017 * Greater than 1 cycle of induction therapy required to achieve remission * Preceding myelodysplastic syndrome (MDS) or myelofibrosis * Therapy-related AML * Presence of FLT3 internal tandem duplications * French-American-British (FAB) M6 or M7 classification * Acute leukemia (ALL or AML) in second (2nd) or greater CR (CR ≥ 2) * Refractory or relapsed AML with =\< 5% bone marrow blasts and no circulating blasts by morphology or proven extramedullary disease * Myelodysplastic syndrome (MDS) with following high risk features: poor cytogenetics (-7, inv(3)/t(3q)/del(3q), del(7q) or complex cytogenetics defined as \>= 3 abnormalities), Revised International Prognostic Scoring System (IPSS-R) risk group intermediate or higher, or treatment-related MDS * Any phase of MDS if patient is \< 21 years of age * Chronic myelogenous leukemia (CML) beyond 1st chronic phase or resistant to tyrosine kinase inhibitors (adults) * Chronic myelomonocytic leukemia (CMML) * Myeloproliferative disorders/myelofibrosis * Hodgkin or non-Hodgkin lymphoma: relapsed chemotherapy-sensitive (complete or partial response) * Female patients must have negative standard pregnancy test (all women of child bearing-potential must have test performed) * Ability to understand and the willingness to sign a written informed consent document * For patients with acute leukemia, CR is defined as =\< 5% marrow blasts by morphology. CR with incomplete count recovery is allowed * DONOR INCLUSION * Unrelated donors matched for human leukocyte antigen (HLA)-A, B, C, DRB1 and DQB1 by high resolution typing * Donors are excluded when preexisting immunoreactivity is identified that would jeopardize donor hematopoietic cell engraftment. This determination is based on the standard practice of the individual institution. The recommended procedure for patients with 10 of 10 HLA allele level (phenotypic) match is to obtain a panel reactive antibody (PRA) screens to class I and class II antigens for all patients before hematopoietic cell transplantation (HCT). If the PRA shows \> 10% activity, then flow cytometric or B and T cell cytotoxic cross matches should be obtained. The donor should be excluded if any of the cytotoxic cross match assays are positive * Only granulocyte colony-stimulating factor (G-CSF) mobilized peripheral blood stem cell (PBSC) will be permitted as a hematopoietic stem cell (HSC) source on this protocol * Donors must meet the selection criteria for administration of G-CSF and apheresis defined based on each institution's standard practice protocol for unrelated donors * Donors must be capable of giving informed consent Exclusion Criteria: * Prior autologous or allogeneic stem cell transplant * Performance status: Karnofsky score \<60 or Lansky score \<50 for patients \<16 years old * Uncontrolled infection. The protocol principal investigator (PI) will be final arbiter if there is uncertainty regarding whether a previous infection is under adequate control to allow enrollment in the study * Positive serology for human immunodeficiency virus (HIV)-1, 2 or human T-lymphotropic virus (HTLV)-1, 2 * Left ventricular ejection fraction \< 45%. Uncontrolled arrhythmias or symptomatic cardiac disease * Symptomatic pulmonary disease. Forced expiratory volume in 1 second (FEV1), forced vital capacity (FVC), diffusion capacity of the lung for carbon monoxide (DLCO) =\< 50% of predicted (corrected for hemoglobin). Use of continuous supplemental oxygen * Calculated (Cockroft-Gault; or appropriate calculation for pediatric patients) serum creatinine clearance \<60 mL/min. If the calculated CrCl is 50-60 mL/min, but a measured CrCl by 24 hour urine collection is \> 60 mL/min, this measurement is acceptable * Total serum bilirubin more than twice upper normal limit * Alanine aminotransferase (ALT) or aspartate aminotransferase (AST) more than 3-fold higher than laboratory upper normal limits * History of allergy or anaphylactic reaction to rabbit protein or to any product excipients * Subjects may not be enrolled in other investigational trials with acute or chronic GVHD as the primary endpoint * DONOR EXCLUSION * Donor who will exclusively donate marrow * Donors who are HIV-positive * Potential donors who for psychological, physiological, or medical reasons cannot tolerate administration of G-CSF or apheresis * Donors who are allergic to filgrastim or Escherichia (E.) coli-derived proteins * Donor-related risks to recipients * Positive anti-donor lymphocytotoxic crossmatch * Pregnant or lactating women * Prior malignancy within the last 5 years