Palbociclib and Dexamethasone in Treating Participants With Relapsed or Refractory B-Cell Acute Lymphoblastic Leukemia

Inclusion Criteria:

* Patients must have histologic evidence of relapsed or refractory B-cell ALL
* Patients must have an Eastern Cooperative Oncology Group (ECOG) performance status of 2 or less
* Philadelphia chromosome positive (Ph+) patients must be refractory to or intolerant of standard tyrosine kinase inhibitor therapy
* Patients must be able to consume oral medication
* Patients must have recovered to =\< grade 1 or stabilized from the toxic effects of any prior chemotherapy (except alopecia)
* Creatinine clearance (CrCL) \>= 60 mL/min/1.73 m\^2 calculated by Cockcroft-Gault
* Total bilirubin \< 1.5 x upper limit of normal (ULN)
* Negative serum or urine pregnancy test for women with child-bearing potential
* Patients must be able to sign consent and be willing and able to comply with scheduled visits, treatment plan, procedures, and laboratory testing

Exclusion Criteria:

* Patients must not have evidence of active central nervous system (CNS) disease
* Patients must not be receiving any chemotherapy agents (except hydroxyurea); intrathecal methotrexate and intrathecal cytarabine are permissible
* Patients must not be receiving growth factors (granulocyte colony-stimulating factor \[G-CSF\], granulocyte-macrophage colony-stimulating factor \[GM-CSF\]), except for erythropoietin
* Patient must not have a concurrent active malignancy for which they are receiving treatment.
* Patients with other severe concurrent disease which in the judgment of the investigator would make the patient inappropriate for entry into this study are ineligible
* Patients must not have received any investigational agents within 30 days of study entry unless they have exceeded 5 terminal half-lives of the previous study drug used for treatment
* Patients must not be pregnant or breastfeeding; pregnancy tests must be obtained for all females of child-bearing potential within 10 days prior to enrollment; males or women of childbearing potential may not participate unless they have agreed to use an effective contraceptive method (defined as hormonal contraceptives, intrauterine devices, surgical contraceptives, or condoms)
* Patients who have uncontrolled infection are not eligible; patients must have any active infections under control; fungal disease must have been adequately treated for at least 2 weeks before study entry; subjects with bacteremia must have documented negative blood cultures prior to study entry
* Patients who are candidates for allogeneic transplantation, have a suitable donor, and are willing to undergo transplantation
* Patients who are eligible for and willing to receive treatment with tisagenlecleucel.