Palbociclib in Combination With Chemotherapy in Pediatric Patients With Relapsed or Refractory Acute Lymphoblastic Leukemia (RELPALL2)

Inclusion Criteria: 1. Participants must have a diagnosis of acute lymphoblastic leukemia and disease meets at least one of the following criteria: 1. relapsed or refractory to chemotherapy as defined by ≥ 5% leukemic blasts in the bone marrow or flow cytometry confirmed leukemic blasts in the peripheral blood 2. relapsed after hematopoietic stem cell transplantation (HSCT) 3. Subjects must have had histologic, morphologic or flow cytometric verification of the malignancy at relapse 2. Prior Treatment: 1. Subjects who relapse while receiving standard ALL maintenance chemotherapy will not be required to have a waiting period before entry onto this study. 2. Subjects who relapse on therapy other than standard ALL maintenance must have fully recovered from the acute toxic effects of all prior anti cancer therapy, defined as resolution of all such toxicities to ≤ Grade 2 or lower per the inclusion/exclusion criteria prior to entering this study. 3. At least 14 days must have elapsed since the completion of cytotoxic therapy, with the exception of standard maintenance therapy and steroids. 4. At least 7 days must have elapsed since completion of therapy with a biologic agent. For agents that have known adverse events occurring beyond 7 days after administration, this period prior to enrollment must be extended beyond the time during which adverse events are known to occur. 5. At least 3 half lives must have elapsed since prior therapy that included a monoclonal antibody with the exception of blinatumomab. Subjects must have been off blinatumomab infusion for at least 7 days and all drug related toxicity must have resolved to Grade 2 or lower as outlined in the inclusion/exclusion criteria. 6. At least 42 days must have elapsed since CAR T cell therapy. 7. At least 90 days have elapsed since bone marrow transplant and participant is off immune suppression for \> 2 weeks, if applicable with no evidence of active GVHD. 8. At least 2 weeks must have elapsed since local XRT (small port); ≥ 3 months must have elapsed if prior cranial or craniospinal XRT was received, if ≥ 50% of the pelvis was irradiated, or if TBI was received; ≥ 6 weeks must have elapsed if other substantial bone marrow irradiation was given. 3. Participants must be \< 25 years of age. 4. Karnofsky or Lansky performance score is \> 50% (corresponding to ECOG Score of \< 2). The Lansky performance score should be used for participants \< 16 years and the Karnofsky performance score for participants ≥ 16 years (see Appendix D). Subjects who are unable to walk because of paralysis, but who are up in a wheelchair, will be considered ambulatory for the purpose of assessing the performance score. 5. Adequate renal function defined as glomerular filtration rate \> 60 mL/min/1.73 m2 or serum creatinine based on age as follows: Max serum creatine (mg/dL) Age (years) Male Female \< 6 months 0.4 0.4 6 months to \< 1 year 0.5 0.5 1 to \< 2 years 0.6 0.6 2 to \< 6 years 0.8 0.8 6 to \< 10 years 1 1 10 to \< 13 years 1.2 1.2 13 to \< 16 years 1.5 1.4 \> 16 years 1.7 1.4 6. Adequate hepatic function defined as 1. Total bilirubin ≤ 2 x upper limit of normal (ULN) for age, and 2. ALT \< 3 x ULN for age, unless elevation is due to leukemic infiltration 7. Adequate cardiac function defined as shortening fraction of \> 27% or ejection fraction \> 45%. 8. Adequate pulmonary function defined as 1. No evidence of dyspnea at rest, no exercise intolerance, and a pulse oximetry \> 94%. 2. No evidence of acute pulmonary infiltrates on chest radiograph 9. Adequate central nervous system (CNS) function defined as 1. Subjects with seizure disorder may be enrolled if on allowed anti convulsants and well controlled. Benzodiazepines and gabapentin are acceptable. 2. CNS toxicity \< Grade 2 10. Adequate peripheral nervous system (PNS) function defined as PNS toxicity \< Grade 2 Exclusion Criteria: 1. Extramedullary disease status: subjects with isolated CNS disease or isolated testicular disease are not eligible. 2. Concurrent chemotherapy or targeted anti cancer agents, other than intrathecal therapy. 3. Subjects who have previously received bortezomib or other proteasome inhibitors that did not have a response while receiving the inhibitor are not eligible. Subjects that responded but had a subsequent relapse are eligible. 4. Subjects who have previously received palbociclib or other CDK4/6 inhibitors are not eligible. 5. Subject with concurrent severe and/or uncontrolled medical conditions that, in the opinion of the investigator, may impair participation in the study or the evaluation of safety and/or efficacy. 6. Subjects that have an active, uncontrolled infection are not eligible. 7. Known HIV infection or active hepatitis B (defined as hepatitis B surface antigen-positive) or C (defined as hepatitis C antibody-positive). 8. Pregnant or lactating (female participant of childbearing potential must have negative serum or urine pregnancy test required within 7 days prior to start of treatment). 9. Male or female participant of reproductive potential must agree to use appropriate methods of contraception for the duration of study treatment and for at least 30 days after last dose of protocol treatment. 10. Cumulative anthracyclines must not be projected to exceed 450 mg/m2 doxorubicin equivalents following completion of treatment on protocol. Therefore, for subjects receiving one course on protocol cumulative anthracyclines cannot exceed 400 mg/m2 doxorubicin equivalents at the time of enrollment (≤ 200 mg/m2 doxorubicin equivalents for subject with prior radiation therapy to the mediastinum). 11. Inability or unwillingness or research participant or legal guardian/representative to give written informed consent.