Fludarabine Phosphate and Total-Body Irradiation Followed by Donor Peripheral Blood Stem Cell Transplant in Treating Patients With Acute Lymphoblastic Leukemia or Chronic Myelogenous Leukemia That Has Responded to Treatment With Imatinib Mesylate, Dasatinib, or Nilotinib

Inclusion Criteria: * Patients =\< 12 years of age must be approved by Fred Hutchinson Cancer Research Center (FHCRC) principal investigator (PI) in advance * Patients with a history of Ph+ ALL or CML-BC who, after receiving imatinib mesylate, (or either dasatinib or nilotinib) have \< 15% blasts on morphologic marrow evaluation; patients with no detectable Ph+ ALL by morphologic or molecular assays (complete remission) will be accepted * An appropriately human leukocyte antigen (HLA) matched related or unrelated donor must be prospectively identified who will be available to donate filgrastim (G-CSF) mobilized stem cells * RELATED DONOR: * Related donor who is HLA genotypically identical at least at one haplotype and may be genotypically or phenotypically identical at the allele level at HLA-A, -B, -C, -DRB1, and -DQB1; * Donor must consent to G-CSR administration and leukapheresis; * Donor must have adequate veins for leukapheresis or agree to placement of central venous catheter (femoral, subclavian) * HLA-MATCHED UNRELATED DONOR: * FHCRC matching allowed will be grades 1.0 to 2.1; unrelated donors who are prospectively matched for HLA-A, B, C, DRB1 and DQB1 by high resolution typing; only a single allele disparity will be allowed for HLA-A, B, or C as defined by high resolution typing * A positive anti-donor cytotoxic crossmatch is an absolute donor exclusion; 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 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; for those patients with an HLA class I allele mismatch, flow cytometric or B and T cell cytotoxic cross matches should be obtained regardless of the PRA results * Patient and donor pairs homozygous at a mismatched allele in the graft rejection vector are considered a two-allele mismatch, i.e., the patient is A\*0101 and the donor is A\*0102, and this type of mismatch is not allowed * Only G-CSF mobilized PBSC only will be permitted as a HSC source on this protocol Exclusion Criteria: * Central nervous system (CNS) involvement with leukemia refractory to intrathecal chemotherapy; prior to HSCT (all patients must receive a diagnostic lumbar puncture (LP) with intrathecal (IT) chemotherapy as per standard practice) * Fertile men or women unwilling to use contraceptive techniques during and for 12 months following treatment * Patients with active non-hematologic malignancies (except non-melanoma skin cancers); this exclusion does not apply to patients with non-hematologic malignancies that do not require therapy * Patients with a history of non-hematologic malignancies (except non-melanoma skin cancers) currently in a complete remission, who are less than 5 years from the time of complete remission, and have a \> 20% risk of disease recurrence; this exclusion does not apply to patients with non-hematologic malignancies that do not require therapy * Females who are pregnant or breastfeeding * Patients who are human immunodeficiency virus (HIV)-positive * Patients with poorly controlled hypertension despite multiple antihypertensives * Adults: Karnofsky score \< 60 * Pediatrics: Lansky play-performance score \< 40 * Patients with cardiac ejection fraction \< 35% (or, if unable to obtain ejection fraction, shortening fraction of \< 26%); ejection fraction is required if age \> 50 years or there is a history of anthracycline exposure or history of cardiac disease; patients with a shortening fraction \< 26% may be enrolled if approved by a cardiologist * Diffusing capacity of the lungs for carbon monoxide (DLCO) \< 30% * Total lung capacity (TLC) \< 30% * Forced expiratory volume in one second (FEV1) \< 30% and/or receiving supplementary continuous oxygen; the FHCRC PI of the study must approve enrollment of all patients with pulmonary nodules * Patients with clinical or laboratory evidence of liver disease would be evaluated for the cause of liver disease, its clinical severity in terms of liver function, and the degree of portal hypertension; patients will be excluded if they are found to have fulminant liver failure, cirrhosis of the liver with evidence of portal hypertension, alcoholic hepatitis, esophageal varices, a history of bleeding esophageal varices, hepatic encephalopathy, uncorrectable hepatic synthetic dysfunction evinced by prolongation of the prothrombin time, ascites related to portal hypertension, bacterial or fungal liver abscess, biliary obstruction, chronic viral hepatitis with total serum bilirubin \> 3 mg/dL, or symptomatic biliary disease * Creatinine levels more than 2.2 X's the upper limit of normal (ULN) at the laboratory where the analysis was performed * Patients with active bacterial or fungal infections unresponsive to medical therapy * For patients receiving dasatinib or nilotinib, baseline corrected QT interval (QTc) (Fridericia's method) prolongation greater than 500 msec * RELATED DONORS: * Identical twin * Infection with HIV * Inability to achieve adequate venous access * Known allergy to G-CSF * Current serious system illness * Bone marrow (BM) donors * HLA-MATCHED UNRELATED DONORS: * BM donors * Donors who are HIV-positive and/or, medical conditions that would result in increased risk for G-CSF mobilization and harvest of PBSC