Inotuzumab Ozogamicin and Blinatumomab With or Without Ponatinib in Treating Patients With Newly Diagnosed, Recurrent, or Refractory CD22-Positive B-Lineage Acute Lymphoblastic Leukemia

Inclusion Criteria: * STEP 0: Submission of bone marrow aspirate and peripheral blood for MRD analysis is mandatory prior to registration; the bone marrow sample should be from the first aspiration (i.e. first pull). Aspirate needle should be redirected if needed to get first pull bone marrow aspirate. It should be initiated as soon as possible after pre-registration. The specimens should be sent to the HEME Biobank. * Lumbar Puncture (Spinal Tap) and Intrathecal Methotrexate: * Patients may receive the day 1 of course IA dose of intrathecal (IT) methotrexate during the prior-to-registration lumbar puncture (or the venous line placement) to avoid a second lumbar puncture. If the dose is administered prior to registration, then systemic chemotherapy must begin within 7 days of this IT chemotherapy. * STEP 1: Morphologic diagnosis of precursor B-cell acute lymphoblastic leukemia (ALL) based on World Health Organization (WHO) criteria. Patients with Burkitt lymphoma/leukemia are not eligible. * STEP 1: CD22-positive disease defined as CD22 expression by \>= 20% of lymphoblasts by local hematopathology evaluation. * STEP 1: Philadelphia chromosome/BCR-ABL1-negative or Philadelphia chromosome/BCR-ABL1-positive B-cell ALL by cytogenetics, fluorescence in situ hybridization (FISH), and/or polymerase chain reaction (PCR). * STEP 1: No active central nervous system (CNS) leukemia (i.e. only CNS-1 disease allowed). Active CNS leukemia is defined as morphologic evidence of lymp…