Inotuzumab Ozogamicin and Chemotherapy in Treating Patients With Leukemia or Lymphoma Undergoing Stem Cell Transplantation

Inclusion Criteria:

* Participants age 12 to 75.
* English and non-English speaking participants are eligible.
* CD22+ lymphoid malignancies including B-ALL
* Eligible to receive a reduced-intensity alloSCT

Participants with:

* Indolent lymphoma participants who failed conventional treatment; or,
* Acute lymphoblastic leukemia (ALL), aggressive lymphoma, indolent lymphoma in transformation, or those who have failed ≥ three small molecule inhibitors
* Donor: HLA compatible (8/8 match) related or matched unrelated donor (HLA-A, B, C, DRB1) or mismatched MUD (7/8 match) or haploidentical
* Performance status of 0 to 2, Lansky ≥ 80 for \< 16 years and Karnofsky ≥ 80 for ≥ 16 years of age.
* Adequate organ function at time of study entry

  1. Creatinine less than or equal to 1.6 mg/dL
  2. Bilirubin less than 1.6 mg/dL
  3. SGPT \< 2 x UL
  4. Ejection fraction \>/= 40%
  5. FEV1, FVC and cDLCO \>/= 40%
* Negative Beta HCG test in a woman with child bearing potential defined as not post-menopausal for 12 months or no previous surgical sterilization) or currently breast-feeding. Pregnancy testing is not required for post-menopausal or surgically sterilized women.

Exclusion Criteria:

* Human immunodeficiency virus (HIV) positive.
* Prior autologous transplant less than 1 year prior to consent.
* Active and uncontrolled disease/infection.
* Unable or unwilling to sign consent.
* Current active hepatic or biliary disease (with exception of Gilbert's syndrome).
* Active hepatitis B or C.
* Recent systemic chemotherapy or radiation within 3 weeks of study entry (intrathecal therapy is allowed).

Standard biological agents such as rituximab, TKIs such as ibrutinib and venetoclax are allowed to be given within 3 days prior to receiving inotuzumab ozogamicin. Blinatumomab is allowed to be given until 1 week prior to Day -13 inotuzumab ozogamicin on study.

* Prior inotuzumab ozogamicin within 3 weeks of study entry.
* Peripheral blast count of greater than 10 K/mL.
* QTcF interval \> 470 ms.
* Participants with cognitive impairments and/or any serious unstable pre-existing medical condition or psychiatric disorder that can interfere with safety or with obtaining informed consent or compliance with study procedures.