Brief Summary: This is a phase I/II clinical trial evaluating the activity of combination chemotherapy with venetoclax and navitoclax in children with relapsed or refractory acute lymphoblastic leukemia or lymphoma (rALL) and assessing the combination dose of venetoclax combinations with either blinatumomab for CD19-postive patients or navitoclax and high-dose cytarabine for CD19-negative patients. Primary Objectives * To compare Minimal Residual Disease (MRD)-negative CR/CRi rate in children with relapsed or refractory acute lymphoblastic leukemia or lymphoma (rALL) following Block 1 therapy with venetoclax and navitoclax based reinduction to historical controls. * To identify the recommended phase 2 combination dose (RP2D) of venetoclax based consolidation in novel combinations with a) high-dose cytarabine and navitoclax or b) blinatumomab. Secondary Objectives * To estimate the tolerability and activity of venetoclax based consolidation in novel combinations with a) high-dose cytarabine and navitoclax or b) blinatumomab. * To describe event-free and overall survival in patients treated with this regimen. Exploratory Objectives * To evaluate MRD-negative CR/CRi rates in each prespecified groups: late first relapse B-ALL; early first relapse and second or greater relapse B-ALL; and relapsed T-ALL. * To identify drug sensitivity patterns in patient samples prior to and after receiving combination therapy and evaluate mechanisms of disease resistance/ escape. * To explore immune subsets during and after this regimen. * Evaluate response to therapy in rare relapse patient subsets. * Explore breakthrough infections in children and young adults with relapsed or refractory ALL

Detailed Description: This is a non-randomized phase I/II clinical trial. In Block I, all patients receiving common therapy evaluating the activity of combination chemotherapy with venetoclax and navitoclax in children with relapsed or refractory acute lymphoblastic leukemia or lymphoma (rALL). In Block 2, a phase 1 rolling six design and phase 2 extension will be used to assess the combination dose of venetoclax combinations with either blinatumomab for CD19-postive patients or navitoclax and high-dose cytarabine for CD19-negative patients. One to six participants can be concurrently enrolled onto a dose level, depending on the number of participants enrolled at the current dose level, the number of participants who have experienced a dose/combination limiting toxicity (DLT) at the current dose level, and the number of participants entered but with tolerability data pending. Dose de-escalation will occur if 2 patients at a dose level experience a targeted toxicity. If 6 patients complete a therapy block and experience 0-1 DLT, that dose will be considered the recommended phase 2 dose (RP2D). New enrollment on an arm (2a/2b) will be paused if further enrollment would result in more than 6 patients being treated on that arm before the RP2D is identified. The primary outcome assessment will be a bone marrow with MRD testing on Day 29 of Block 1 therapy. Following that assessment, patients should continue with protocol therapy including either Block 2A (which includes high-dose cytarabine with venetoclax and navitoclax) for patients with CD19 negative leukemia, those with isolated extramedullary relapse, or whose treating physician determines that blinatumomab therapy is not in the patient's best interest; or Block 2B for patients with leukemia which is CD19 positive. Following recovery from Block 2 therapy, all patients except those with late first relapse of B-ALL (≥ 36 months from diagnosis) who are MRD-negative at \<0.01% after Block 1 therapy are off therapy. Further therapy for these patients is at the treating physician's discretion. Patients with late first relapse B-ALL relapse and who are MRD negative will proceed with intensification, interim continuation, and continuation therapy. Patients with Down's Syndrome with CD19 negative leukemia will be off therapy after Block 1 as they are ineligible for Block 2A. Patients in exploratory cohorts I and N who are late first relapse B-ALL and who are MRD-negative after Block 1 therapy may continue on protocol therapy after Block 2 or receive alternative therapy at their treating physician's discretion. Patients in exploratory cohort M and O are off therapy after Block 2. Intervention: Block 1 Therapy: * Venetoclax 120mg/m\^2 (max 200mg) PO, day 1; 240mg/m\^2/dose (max 400mg) PO, 21 doses, days 2-22 * Navitoclax 25mg PO (for patients 20 - \<45kg) OR 50mg PO (for patients \>=45kg), days 3-22 * Dexamethasone 5mg/m\^2/dose BID PO/IV, days 1-7 and 15-22 * Vincristine 1.5mg/m\^2/dose (max 2mg) IV, days 1, 8, 15, and 22 * Calaspargase pegol 2500units/m\^2/dose IV, day 2 * Dasatinib 80mg/m\^2/day (max 140mg) PO, days 1-28 for ABL-class fusions and non-ETP T-ALL * IT MHA: Intrathecal (IT) MHA (methotrexate 12 mg/hydrocortisone 24mg/cytarabine 36mg), days -7 to 1. Continue intrathecal chemotherapy weekly until negative x 2. * Leucovorin 5mg/m2 (max 5mg) PO or IV (24 and 30 hours after each LP). Calaspargase may be replaced by alternative forms of asparaginase due to local practice or prior allergy. If pegaspargase is used, two doses (on days 2 and 15) of 2500units/ m\^2 replace the single day 2 dose of calaspargase. For patients in exploratory cohort N, asparaginase will be omitted. For patients in exploratory cohort O, the Day 15 dose of pegaspargase may be omitted and the Day 2 dose may be capped at 2000units/m2 with a maximum dose of 3750 units. Testicular Radiation: Patients with persistent testicular involvement by leukemia at the end of Block 1 may continue on study and may receive testicular radiation during Block 2 therapy. Block 2a - For patients with ≥ 5% of leukemic blasts which do not have detectable CD19 on their surface, those with isolated extramedullary relapse, or whose physician determines this therapy arm is in their patient's best interest. Patients with Down Syndrome are not eligible for Block 2a therapy. Patients in exploratory cohort O are not eligible for the phase I/dose de-escalation phase of the study but may enroll on block 2 therapy after the dose has been established.: * Venetoclax 240mg/m\^2 (max 400mg) PO, days 1-7 * Navitoclax 25mg (20-\<45kg) OR 50mg (\>=45kg) PO, days 1-7 * Dexamethasone 3mg/m\^2/dose PO/IV BID, days 1-5 * Calaspargase pegol 1000units/m\^2 IV, day 3 * Dasatinib 80mg/m2/day (max 140mg) PO, days 1-28 for ABL-class fusion and non-ETP T-ALL/ALLy * IT MHA: Methotrexate 12 mg IT, Hydrocortisone 24mg IT, Cytarabine 36mg IT, one dose with end of cycle bone marrow assessment * Cytarabine dosing during phase I portion for Block 2a: * Dose level 1: 3000mg/m\^2/dose every 12hours(h) IV infusion over 3 hours, days 1-2 * Dose level 0: 1500mg/m\^2/dose every 12h IV infusion over 3 hours, days 1-2 * Dose level -1: 1000mg/m\^2/dose every 12h IV infusion over 3 hours, day 1 Calaspargase may be replaced by alternative forms of asparaginase. For patients in exploratory cohort N, asparaginase will be omitted. Block 2b Therapy: For patients with \>95% of leukemic blasts with detectable CD19 on their surface. Patients with isolated extramedullary relapse are ineligible for Block 2b therapy. Patients in exploratory cohort O are not eligible for the phase I/ dose de-escalation phase of the study but may enroll on block 2 therapy after the dose has been established. * Blinatumomab; for patients with end of Block 1 MRD \>5%, 5mcg/m\^2/day (max 9mcg/day; patients weighing ≥45kg receive 9mcg/day fixed dose) IV, days 1-7 * Blinatumomab; for patients with end of Block 1 MRD \>5%, 15mcg/m\^2/day (max 28mcg/day; patients weighing ≥45kg receive 28mcg/day fixed dose) IV, days 8-28 * Blinatumomab; for patients with end of Block 1 MRD ≤5%, 15mcg/m\^2/day (max 28mcg/day; patients weighing ≥45kg receive 28mcg/day fixed dose) IV, days 1-28, * Dexamethasone 10mg/m\^2 (max 20mg) 30-60 minutes prior to day 1 blinatumomab, PO/IV, days 1, 8 (For patients who start blinatumomab at 5mcg/m2/day, an additional dose of dexamethasone will be given on day 8 30-60 minutes prior to dose increase. For patients whose blinatumomab is interrupted for more than 4 hours for any reason, an additional dose of dexamethasone will be given prior to restarting blinatumomab.) * Dasatinib 80mg/m\^2/day (max 140mg) PO, days 1-28 for ABL-class fusions * IT MHA: Methotrexate 12 mg IT, Hydrocortisone 24mg IT, Cytarabine 36mg IT, day 29 with end of cycle bone marrow * Venetoclax dosing during phase I portion for Block 2b: * Dose level 1: 240mg/m\^2 (max 400 mg) PO, days 8-28 * Dose level 0: 240mg/m\^2 (max 400 mg) PO, days 8-21 * Dose level -1: 240mg/m\^2 (max 400 mg) PO, days 8-14 Intensification Therapy - For late first relapse B-ALL and MRD\<0.01% after Block 1 only. Patients in exploratory cohort O are excluded.: * High Dose (HD) Methotrexate 5000mg/m\^2/dose IV: 500mg/m\^2 over 60 minutes then 4500mg/m\^2 over 23 hours; or targeted 65 microM, day 1 and 15 * Mercaptopurine 50mg/m\^2/dose PO daily, days 1-28 * IT MHA: Methotrexate 12 mg IT, Hydrocortisone 24mg IT, Cytarabine 36mg IT, D15 * Leucovorin 15mg/m2/dose PO/IV every 6 hours (start 42 hours after HD-MTX begins) D2 and 16 (at least 3 doses). Interim Continuation Therapy 1 and 2 (8 weeks each) - For late first relapse B-ALL and MRD\<0.01% after Block 1 only: * Dexamethasone 3mg/m\^2/dose BID PO/IV, days 1-5 * Vincristine1.5mg/m\^2(max 2mg) IV, day 1 * Mercaptopurine 75mg/m\^2/dose PO daily or 60mg/m2/day for patients with known heterozygous inactivating mutations of TPMT or NUDT15, days 1-49 * Methotrexate 20mg/m\^2/dose PO/IV, days 8, 15, 29, 36 * High Dose (HD)-Methotrexate 5000mg/m\^2/dose IV: 500mg/m2 over 60 minutes then 4500mg/m\^2 over 23 hours; or targeted 65 microM, day 22 * Leucovorin 15mg/m\^2/dose PO/IV every 6 hours: Start 42 hours after HD-MTX, days 24 and 25 (at least 3 doses) * Cyclophosphamide 300mg/m\^2/dose IV over 15-30 minutes, days 43 and 50 * Etoposide 150mg/m\^2/dose IV over 1-2 hours, days 43 and 50 * Cytarabine 50mg/m\^2/dose IV over 1-30 minutes, days 44-47 and 51-54 * Dasatinib 80mg/m\^2/day PO (Max 140mg), days 1-56 for ABL-class fusions * IT MHA: Methotrexate 12 mg IT, Hydrocortisone 24mg IT, Cytarabine 36mg IT, days 22 and 43 Following recovery from Interim Continuation therapy 1, patients will repeat Block 2 therapy according to their initial Block assignment (2A for CD19-negative, 2B for CD19-positive) and receive a second cycle of interim continuation. Patients will not receive venetoclax or navitoclax during this repeated Block 2 cycle. Following recovery from Interim Continuation 2, patients will proceed to Continuation therapy. Continuation Therapy - For late first relapse B-ALL and MRD \<0.01% after Block 1 only. Continuation cycles last 8 weeks (56 days) each. Continuation therapy continues until 2 years from the start of protocol therapy.: * Dexamethasone 6mg/m\^2/dose BID PO/IV, days 1-5, 29-33 * Vincristine 1.5mg/m\^2 (max 2mg) IV, day 1 and 29 * Mercaptopurine 75mg/m\^2/dose PO daily or 60mg/m\^2/day for patients with known heterozygous inactivating mutations of TPMT or NUDT15 days 1-56 * Methotrexate 20mg/m\^2/dose PO/IV days 8, 15, 22, 36, 43 and 50 * Dasatinib 80mg/m\^2/day PO (Max 140mg) days 1-56 for ABL-class fusions * IT MHA: Methotrexate 12 mg IT, Hydrocortisone 24mg IT, Cytarabine 36mg IT day 1 and 29 (patients with CNS2 or traumatic lumbar puncture with blasts present at the time of relapse will receive day 1 and 29 IT MHA for the first year of continuation therapy. Following this, patients will receive IT MHA on day 1 of each cycle until the completion of therapy. Patients with CNS3 disease at relapse will receive day 1 and 29 IT MHA during cycles 1 and 2 of continuation and then proceed to chemoradiation.) Chemoradiation for those with CNS3 disease at relapse (late first relapse \[≥36 months from diagnosis\] B-ALL and MRD\<0.01% after Block 1 only): Patients with CNS3 disease at relapse will receive 18Gy cranial irradiation after Cycle 2 of Continuation Therapy. Patients will not receive additional intrathecal chemotherapy following irradiation. Chemoradiation lasts 21 days. * Dexamethasone 6mg/m2/dose BID PO/IV days 1-5 and 15-19 * Vincristine 1.5mg/m\^2 (max 2mg) IV days 1, 8 and 15 * Calaspargase 2500units/m\^2/dose IV, Day 2 (Calaspargase may be replaced by alternative forms of asparaginase and will be omitted for patients in exploratory cohort N) * Dasatinib 80mg/m\^2/day PO (Max 140mg) days 1-21 for ABL-class fusions