Inclusion Criteria:

          -  Diagnosis of B-ALL with no evidence of minimal residual disease in the bone marrow by
             multi-parameter flow cytometry (FC-MRD negative, <0.01%) and meet at least one of the
             following:

               1. In remission after first relapse or greater (≥ CR2)

               2. Very-high risk biology ALL that is proceeding to HCT in first remission (e.g.
                  Induction failure, Severe-hypodiploidy, Ph-like ALL)

               3. First remission with persistent disease identified as end of consolidation (EOC)
                  MRD > 0.01%.

          -  Patients must have an available unrelated or haploidentical donor

          -  Age ≤ 25 years at time of study enrollment

          -  Karnofsky Performance Status ≥ 60% for patients 16 years and older and Lansky Play
             Score ≥ 60 for patients under 16 years of age

          -  Have acceptable organ function as defined within 14 days of study registration: Renal:
             creatinine clearance or radioisotope GFR ≥ 60 mL/min/1.73m2 Hepatic: ALT < 5 x upper
             limit of normal (ULN) and total bilirubin ≤ 3 mg/dL Cardiac: left ventricular ejection
             fraction ≥ 40% by ECHO/MUGA Pulmonary: No evidence of dyspnea at rest. No supplemental
             oxygen requirement. If measured, carbon monoxide diffusion capacity (DLCO) > 50%.
             Central Nervous System: Based on clinical exam, no concern for/evidence of active CNS
             infection. Patients with fully treated prior CNS infections are eligible. Patients
             with seizure disorders may be enrolled if seizures are well-controlled on
             anticonvulsant therapy.

          -  Patients who have experienced their relapse after HCT are eligible, provided they have
             no evidence of acute or chronic Graft-versus-Host Disease (GVHD) and are off all
             transplant immune suppression therapy for at least 7-days (e.g. steroids,
             cyclosporine, tacrolimus). Steroid therapy for non-GVHD and/or non-leukemia therapy is
             acceptable.

          -  Immunotherapy: At least 42 days after the completion of any type of immunotherapy
             aside from blinatumomab (e.g. tumor vaccines or CAR T-cell therapy).

          -  XRT: Cranial or craniospinal XRT is prohibited during protocol therapy. ≥ 90 days must
             have elapsed if prior TBI, cranial or craniospinal XRT

          -  Sexually active females of child bearing potential must agree to use adequate
             contraception (diaphragm, birth control pills, injections, intrauterine device [IUD],
             surgical sterilization, subcutaneous implants, or abstinence, etc.) for the duration
             of treatment and for 2 months after the completion of blinatumomab therapy. Sexually
             active men must agree to use barrier contraceptive for the duration of treatment and
             for 2 months after the completion of blinatumomab therapy.

          -  Voluntary written consent before performance of any study-related procedure not part
             of normal medical care, with the understanding that consent may be withdrawn by the
             subject at any time without prejudice to future medical care.

          -  All patients enrolled in this study must have been enrolled in the Blinatumomab
             Bridging Therapy (BBT) Trial

        Exclusion Criteria:

          -  Active extramedullary disease or presence of chloromatous disease.

          -  Receiving concomitant chemotherapy, radiation therapy; immunotherapy or other
             anti-cancer therapy for treatment of disease other than is specified in the protocol.

          -  Systemic fungal, bacterial, viral, or other infection not controlled (defined as
             exhibiting ongoing signs/symptoms related to the infection and without improvement,
             despite appropriate antibiotics or other treatment). Patients with possible fungal
             infections must have had at least 2 weeks of appropriate anti-fungal antibiotics and
             be asymptomatic.

          -  Pregnant or lactating. The agents used in this study are known to be teratogenic to a
             fetus and there is no information on the excretion of agents into breast milk. All
             females of childbearing potential must have a blood test or urine study within 7 days
             prior to registration to rule out pregnancy.

          -  Known allergy to any chemotherapies or targeted agents included in this protocol.

          -  Participating in a concomitant Phase 1 or 2 study involving treatment of disease.

          -  Active malignancy other than B-ALL.