Study of Palifermin (Kepivance) in Persons Undergoing Unrelated Donor Allogeneic Hematopoietic Cell Transplantation

* INCLUSION CRITERIA: Patients meeting below eligibility criteria are eligible to receive suitable disease specific therapy for the purposes of disease control while the donor search takes place * The patient is greater than or equal to 18 years of age. * Ability of subject to understand and the willingness to sign a written informed consent document. * Karnofsky performance score \>= 60. * No suitable HLA matched sibling donor is available, and the patient has one or more potentially suitable HLA matched unrelated donor(s) in the National Marrow Donor Registry or other available registry. * The evaluation of donors shall be in accordance with existing National Marrow Donor Program (NMDP) Standard Policies and Procedures * HLA-matched donors are defined by allele matching at HLA-A, -B, -C, * There is a high likelihood that the patient, in the opinion of the principal investigator (PI) or lead associate investigator (LAI), will meet the research phase eligibility criteria and proceed to transplant after induction phase therapy is completed. * Diagnosis of hematologic malignancy meeting at least one of the disease status criteria outlined in the table below. Diagnoses must be confirmed by the National Cancer Institute (NCI) laboratory of pathology. * Recipients with acute myeloid leukemia (AML) in first complete remission (CR1) must have one of the following: * Adverse cytogenetics (as evaluated by history) as defined as complex karyotype (\> 3 abnormalities); inv(3) or t(3;3); t(6;9); t(6;11); monosomy 7; trisomy 8, alone or with an abnormality other than t(8;21), t(9;11), inv(16) or t(16;16); or t(11;19)(q23;p13.1) or adverse-risk per European LeukemiaNet (ELN) 2017 criteria. * Intermediate-risk disease, such as cytogenetically normal AML (CN-AML) with mutations in FMS-like tyrosine kinase 3 (FLT3), DNA methyl transferase 3A (DMNT3A), or additional sex coombs like 1 (ASXL1) or per ELN 2017 criteria. * Primary induction failure, defined as failure to achieve CR with primary induction chemotherapy. * Secondary AML, defined as AML related to antecedent myeloid neoplasm or cytotoxic chemotherapy. * Hyperleukocytosis, white blood cell (WBC) \> 100,000, at diagnosis. * Recipients with ALL in CR1 must have one of the following: * Adverse cytogenetics defined as translocations involving t(4;11), t(1;19), t(8;14), 11q23, t(9;22) or bcr-abl rearrangement, Philadelphia chromosomelike (Ph-like ALL), or complex cytogenetic abnormalities. * Presence of minimal residual disease using multicolor flow cytometry or other analytic technique after primary induction chemotherapy. * Primary induction failure, defined as failure to achieve complete remission (CR) with primary induction chemotherapy. * Recipients with myelofibrosis must have at least 2 of the following features, or be Dynamic International Prognostic Scoring System (DIPSS) intermediate-2 or high risk: * Hemoglobin \< 10 g/dl, or \> 10 g/dl with transfusion dependence. * WBC \< 4,000 or \> 30,000/mm\^3 or requires cytoreductive therapy to maintain WBC \< 30,000/mm\^3. * Abnormal cytogenetics. * Patients with lymphoma must ideally have at least stable disease from last therapy however if the PI or LAI believes there is a high likelihood of response to induction chemotherapy (EPOCH-F+/R), then the patient may be enrolled on the induction phase arm. For enrollment on the research phase arm, the patient must have at lease stable disease which is defined as: * Absence of disease progression for at least 8 weeks after previous therapy or 12 weeks after autologous transplantation. * Patients who are less than 8 weeks from previous therapy or 12 weeks from autologous transplantation may participate in the study at the discretion of the PI or LAI as long as they do not have progressive disease. * Multiple myeloma in complete remission is defined as per Dr. Brian G.M. Durie (Durie BG) et al. * Negative immunofixation on the serum and urine and disappearance of any soft tissue plasmacytomas and \<=5% plasma cells in the bone marrow. CR requires two consecutive assessments by serum and urine immunofixation made at any time prior to enrollment. CR also requires no known evidence of progressive or new bone lesions if radiographic studies are performed. Confirmation with repeat bone marrow is not needed. * The recipient has acceptable end organ function as defined by: * Diffusing capacity for carbon monoxide (DLCO) \> 50% of the expected value (using United States of America (USA)-Information Technology Service (ITS)-National Institutes of Health (NIH) equation) when corrected for Hgb (DLCO Adj.) * 24hr creatinine clearance or calculated (using the Cockcroft-Gault formula) creatinine clearance \> 60 ml/min/1.73 (induction phase only) * Left ventricular ejection fraction \> 45% * Serum total bilirubin less than 2.5 mg/dl, and serum alanine aminotransferase (ALT) and aspartate transferase (AST) values less than or equal to 2.5 times the upper limit of normal. Patients with elevations of serum total bilirubin up to 10 mg/dl and/or ALT or AST up to 10 times the upper limit of normal may be considered for participation if such elevations are thought to be due to liver involvement by malignancy. However, in these latter patients, if the bilirubin (BR) level does not decrease to less than or equal to 2.5 mg/dl, or AST/ALT do not decrease to less than or equal to 2.5 times the upper limit of normal after induction chemotherapy, eligibility for the transplant (research) phase will be at the discretion of the principal investigator (PI). * Patients who are hepatitis B core antibody positive and or have positive hepatitis B surface antigen will require hepatology consultation. The risk/benefit profile of transplant and hepatitis B will be discussed with the patient and eligibility determined by the PI or the LAI. * Patient may have a hepatitis C infection. However, each patient will require a hepatology consultation. The risk/benefit profile of transplant and hepatitis C will be discussed with the patient and eligibility determined by the PI or the LAI. * Palifermin has had embryotoxic and fetotoxic effects in animal studies. For this reason and because the other therapeutic agents used in this trial are known to be teratogenic, women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and for the duration of active study therapy. Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately. Research Phase Inclusion Criteria: Verification of donor eligibility (clearance must be received from the NMDP) * Donors are evaluated by NMDP affiliated donor centers per NMDP Standards. * Donors who are medically suitable, but ineligible by Food and Drug Administration (FDA) guidelines may still donate peripheral blood stem cells (PBSC) with documentation of urgent medical need by the PI. * Patients who receive stem cell products from ineligible donors will be informed of any increase in risk of transfusion-related diseases prior to initiation of conditioning chemotherapy. * Donors who are ineligible or unwilling to donate bone marrow will not be eligible to donate to study recipients. However, in the event that the patient has already begun conditioning chemotherapy and a donor PBSC collection is terminated early for donor-related medical concerns, a bone marrow graft may be infused. Should this occur, the recipient will continue to be managed on this protocol for all transplant-related care and complications. Patient will stay on study, but clinical outcomes will not be eligible for the statistics and end point calculations. * Inadequate stem cell collection from the selected donor is defined as less than or equal to 2 x 10\^6 cluster of differentiation 34 (CD34+) cells/kg. In most cases, donor cell collections are infused fresh. If a fresh collection is found to have an inadequate cell count, the cells will still be infused, but the recipient will be removed from the study and managed clinically for all transplant related care and complications on this protocol. If the patient fails to engraft, the donor may be requested for a second collection or an emergency bone marrow harvest at the discretion of the PI and NMDP Medical Director. In the event of an inadequate collection obtained prior to patient conditioning, the donor may be asked to donate a second time, or another eligible donor may be requested. * Renal and hepatic function continues to meet eligibility criteria, reassessed as follows: * 24-hour creatinine clearance or calculated (using the Cockcroft-Gault formula) creatinine clearance \> 60 mL/min/1.73 m\^2 (induction phase only) * (((140-age)\*mass(kg))/(72\*serum creatinine (mg/dL))) x 1.73 m\^2/patients body surface area (BSA) * If the patient is female, multiply the above by 0.85 * In patients with suspected liver disease, bilirubin must be less than or equal to 2.5 mg/dL, AST and ALT must be less than or equal to 2.5 times institutional upper limit of normal (ULN) * The malignancy must be restaged prior to research phase and must not have progressed during induction chemotherapy (stable disease or better). Persons with acute leukemia, myelodysplastic syndrome (MDS)/refractory anemia with excess blasts (RAEB)-I or II or chronic myeloid leukemia (CML) with previous accelerated or blast phase must have \<5% blasts in the bone marrow. Persons with chronic phase CML may have up to 10% blasts in the bone marrow. EXCLUSION CRITERIA (applies to all phases of this protocol): * Active infection that is not responding to antimicrobial therapy. * Active central nervous system (CNS) involvement by malignancy (patients with known positive cerebrospinal fluid (CSF) cytology or parenchymal lesions visible by computed tomography (CT) or magnetic resonance imaging (MRI). * Previous other malignancies unless they have undergone curative intent therapy for that malignancy and (1) have had no evidence of that disease for 5 years, and/or (2) be deemed at low risk for recurrence (less than or equal to 20% at 5 years). * Human immunodeficiency virus (HIV) positive patients are ineligible as allogeneic stem cell transplant is not yet a proven approach in this patient population, and patients are at increased risk of lethal infections when treated with marrow suppressive therapy. * Pregnant women are excluded from this study because palifermin has been shown to be embryotoxic and fetotoxic in animal studies. Because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with palifermin, breastfeeding should be discontinued for the duration of active study therapy. These potential risks may also apply to other agents used in this study. * History of psychiatric disorder or any other condition which may compromise compliance with transplant protocol or expose patient to unnecessary risk as determined by principal investigator or lead associate investigator.