Inclusion Criteria: * Acute leukemia (AL) that includes acute myeloid leukemia (AML) / acute lymphoblastic leukemia (ALL) / mixed phenotype leukemia (MPAL) in complete morphological remission (CR) with or without detectable minimal residual disease (MRD); complete morphological remission is defined by the presence of less than 5% of detectable blasts in bone marrow specimen, evaluated per standard of care. Patients with documented CR but without hematologic recovery since last chemotherapy are considered eligible to the study * Chronic myelogenous leukemia (CML), except refractory blast crisis. To be eligible in first chronic phase, patients must have failed or be intolerant to at least one tyrosine-kinase inhibitor * Chronic myelomonocytic leukemia (CMML) * Myelodysplastic syndromes (MDS) * Lymphoblastic, Burkitt's and other high-grade lymphoma in any complete (CR) or partial (PR) response * CR and PR are defined according to Lugano classification: Recommendations for Initial Evaluation, Staging, and Response Assessment of Hodgkin and Non-Hodgkin Lymphoma: The Lugano Classification * Low grade lymphoma (chronic lymphocytic leukemia \[CLL\]/small lymphocytic lymphoma \[SLL\], marginal zone lymphoma, follicular lymphoma) progressed after two treatment regimens, in CR/PR * For CLL/SLL, CR and PR are defined according to: International Workshop on CLL (iwCLL) guidelines for diagnosis, indications for treatment, response assessment, and supportive management of CLL * CR and PR are defined according to Lugano classification: Recommendations for Initial Evaluation, Staging, and Response Assessment of Hodgkin and Non-Hodgkin Lymphoma: The Lugano Classification * Large cell lymphoma in \> second CR (CR2)/ \>= PR2 * CR and PR are defined according to Lugano classification: Recommendations for Initial Evaluation, Staging, and Response Assessment of Hodgkin and Non-Hodgkin Lymphoma: The Lugano Classification * Mantle cell lymphoma, lymphoplasmacytic lymphoma and prolymphocytic leukemia may be eligible after initial therapy if in CR/PR * CR and PR are defined according to Lugano classification: Recommendations for Initial Evaluation, Staging, and Response Assessment of Hodgkin and Non-Hodgkin Lymphoma: The Lugano Classification * For prolymphocytic leukemia (PLL), CR is defined as a normalization of lymphadenopathies (long-axis diameter \< 1 cm) and splenomegaly (\< 13 cm), absence of constitutional symptoms, PLL cells \< 5% in bone marrow and circulating lymphocytes count \< 4 x 10\^9/L. Patients without hematopoietic recovery are considered eligible to the study. PR is defined as a decrease of \>= 30% of the sum of lymphadenopathies' long-axis diameters, a decrease of \>= 50% in spleen vertical length beyond normal from baseline, peripheral blood (PB) lymphocytes =\< 30 x 10\^9/L (and a decrease of \>= 50% from baseline) * Hodgkin Lymphoma in \> CR2/PR2 * CR and PR are defined according to Lugano classification: Recommendations for Initial Evaluation, Staging, and Response Assessment of Hodgkin and Non-Hodgkin Lymphoma: The Lugano Classification * Subjects must be \>= 6 months old * Karnofsky \>= 70 or Eastern Cooperative Oncology Group (ECOG) 0-1 (for adults) * Lansky score \>= 50 (for children) * Adequate cardiac function defined as absence of decompensated congestive heart failure or uncontrolled arrhythmia AND left ventricular ejection fraction \>= 40% or shortening fraction \> 22% * Adequate pulmonary function defined as absence of oxygen (O2) requirements and one of the following: * Diffusion capacity of the lung for carbon monoxide (DLCO) corrected \>= 70% mm Hg * DLCO corrected between 60% - 69% mm Hg and partial pressure of oxygen (pO2) \>= 70 mm Hg * DLCO corrected between 50% - 59% mm Hg and pO2 \>= 80 mm Hg Pediatric patients unable to perform pulmonary function tests must have O2 saturation \>= 92% on room air. May not be on supplemental oxygen * Total bilirubin \< 2 x upper limit of normal (ULN) unless felt to be related to Gilbert's disease or hemolysis * Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) =\< 2.5 x ULN * Alkaline phosphatase =\< 5 x ULN * Creatinine \< 2.0 mg/dl (adults) or estimated creatinine clearance \> 40 ml/min (pediatrics) * All adults with a creatinine \> 1.2 or a history of renal dysfunction must have estimated creatinine clearance \> 40 ml/min * If recent mold infection, e.g., aspergillus, must be cleared by infectious disease to proceed * Patients who have undergone prior allogeneic hematopoietic cell transplant are eligible, but the prior transplant must have been performed at least 3 months prior to enrollment, unless in case of graft failure from the prior transplant * Written and signed informed consent * DONOR: Donors must be haploidentical relatives of the patients. Donor-recipient compatibility will be tested through HLA typing at high resolution for the HLA loci (-A, -B, -C, -DRB1, -DQB1). Donor and recipient should share at least 5/10 HLA loci * DONOR: Age \>= 12 years * DONOR: Weight \>= 40 Kg * DONOR: Ability of donors younger than 18 years of age to undergo apheresis without use of a vascular access device. Vein check must be performed and verified by an apheresis nurse prior to arrival. * DONOR: Donor must meet selection criteria as defined by the Foundation of the Accreditation of Cell Therapy (FACT) and will be screened per the American Association of Blood Banks (AABB) guidelines * DONOR: In case of more available haploidentical donors, selection criteria should include, in this order: * For cytomegalovirus (CMV) seronegative recipients, a CMV seronegative donor * Red blood cell compatibility * Red blood cell (RBC) cross match compatible * Minor ABO incompatibility * Major ABO incompatibility Exclusion Criteria: * Active, uncontrolled, life-threatening viral, bacterial or fungal infection requiring treatment at time of conditioning regiment administration and transplantation * Presence of a malignancy other than the one for which the transplant is being performed, with an expected survival less than 75% at 5 years * Pregnant or breastfeeding * Known hypersensitivity to treosulfan, fludarabine or cyclophosphamide * Dosing with another investigational agent within 30 days prior to entry in the study * Central nervous system (CNS) leukemic involvement not clearing with intrathecal chemotherapy and/or cranial radiation prior to initiation of conditioning (day -6) * DONOR: Since detection of anti-donor-specific-antigen antibodies (anti-DSA) is associated with higher graft rejection rate, patients will be screened for anti-DSA pre-transplant. Patients with DSA mean fluorescent intensity (MFI) \< 5000 after desensitization treatment, will be considered eligible to participate in the study. The first 10 subjects enrolled in the trial will be DSA-negative.