Flotetuzumab in Treating Patients With Recurrent or Refractory CD123 Positive Blood Cancer

Inclusion Criteria: * Documented informed consent of the participant and/or legally authorized representative. * Assent, when appropriate, will be obtained per institutional guidelines * Agreement to allow the use of archival tissue from diagnostic tumor biopsies * If unavailable, exceptions may be granted with study principal investigator (PI) approval. * Eastern Cooperative Oncology Group (ECOG) =\< 2 * Histologically confirmed diagnosis of * Cohort A. Acute lymphoblastic leukemia * B-cell phenotype: patients with relapsed or refractory ALL who have received at least 2 prior regimens and failed or are ineligible for CD19-based target therapy * T-cell phenotype: patients with relapsed or refractory who have received at least 1 prior regimen * Cohort B. Other CD123+ hematological malignancies that failed standard regimens, excluding acute myeloid leukemia and myelodysplastic syndrome * Blastic plasmacytoid dendritic cell neoplasm (BPDCN) patients who have failed or relapsed after initial therapy * Chronic myeloid leukemia (CML) patients who have failed or relapsed or ineligible for third generation tyrosine kinase inhibitor (ponatinib) * Hairy cell leukemia patients who have failed or progressed shortly after purine analogs or failed 2 cycles of purine analog * Systemic mastocytosis patients who have failed or progressed on midostaurin * Hodgkin lymphoma patients who have failed or relapsed after PD-1/PD-L1- inhibitors and brentuximab vedotin * Advanced acute leukemia patients with ambiguous lineage or biphenotypic leukemia that failed 2 lines of prior regimens * Patients with any other advanced CD123+ hematological malignancy who have failed standard therapy per the treating physician's judgement * Relapsed or refractory disease as defined above * Tumor cells expressing CD123 either by flow cytometry or immunohistochemistry staining as defined below * Measurable disease of at least 1.5 cm on computed tomography (CT)/magnetic resonance imaging (MRI) for cases without bone marrow involvement * Peripheral blast count \< 20,000/ul at the time of initiation of infusion on cycle 1 day 1 * Life expectancy of at least 4 weeks * Fully recovered from the acute toxic effects (except alopecia) to =\< grade 1 to prior anti-cancer therapy * Absolute neutrophil count (ANC) \>= 750/ul (to be performed within14 days prior to day 1 of protocol therapy unless otherwise stated) * NOTE: Growth factor is not permitted within 14 days of ANC assessment unless cytopenia is secondary to disease involvement * Platelets \>= 50,000/ul (to be performed within14 days prior to day 1 of protocol therapy unless otherwise stated) * NOTE: Platelet transfusions are not permitted within 14 days of platelet assessment unless cytopenia is secondary to disease involvement * Lumbar puncture to assess presence of central nervous system (CNS) disease if there are symptoms and signs concerning for CNS involvement (to be performed within14 days prior to day 1 of protocol therapy unless otherwise stated) * Total bilirubin =\< 1.5 X upper limit of normal (ULN) (unless has Gilbert's disease) (to be performed within14 days prior to day 1 of protocol therapy unless otherwise stated) * Aspartate aminotransferase (AST) =\< 2.5 x ULN (to be performed within14 days prior to day 1 of protocol therapy unless otherwise stated) * Alanine aminotransferase (ALT) =\< 2.5 x ULN (to be performed within14 days prior to day 1 of protocol therapy unless otherwise stated) * Serum creatinine level =\< 1.5 times the ULN or a calculated or measured creatinine clearance of \>= 50 mL/min per 24 hour urine test or the Cockcroft-Gault formula (to be performed within14 days prior to day 1 of protocol therapy unless otherwise stated) * Seronegative for human immunodeficiency virus (HIV) antigen (Ag)/antibody (Ab) combo, hepatitis C virus (HCV)\*, active hepatitis B virus (HBV) (surface antigen negative), and syphilis (rapid plasma reagin \[RPR\]) (to be performed within14 days prior to day 1 of protocol therapy unless otherwise stated) * If positive, hepatitis C ribonucleic acid (RNA) quantitation must be performed * Meets other institutional and federal requirements for infectious disease titer requirements * Note Infectious disease testing to be performed within 28 days prior to day 1 of protocol therapy * Women of childbearing potential (WOCBP): negative urine or serum pregnancy test (to be performed within14 days prior to day 1 of protocol therapy unless otherwise stated) * If the urine test is positive or cannot be confirmed as negative, a serum pregnancy test will be required * Agreement by females and males of childbearing potential\* to use an effective method of birth control or abstain from heterosexual activity for the course of the study through at least 6 months after the last dose of protocol therapy. * Childbearing potential defined as not being surgically sterilized (men and women) or have not been free from menses for \> 1 year (women only) Exclusion Criteria: * Autologous or allogeneic hematopoietic cell transplant performed within 100 days prior to study drug administration in day 1 of cycle 1 of protocol therapy. However, patients who received allogeneic hematopoietic cell transplantation (HCT) more than 100 days are allowed if no active graft versus host disease (GVHD) \> grade 1 and not actively on systemic immunosuppressive therapy * Chemotherapy, radiation therapy, biological therapy, within 14 days prior to day 1 of protocol therapy. Maintenance-type ALL chemotherapies, including vincristine and mercaptopurine are allowed up to 7 days before starting therapy. High dose steroids are allowed up to 3 days before starting therapy. Cytoreduction with hydroxyurea is allowed to control leukocytosis until to the day of starting therapy. Hydroxyurea can be given during cycle 1 of flotetuzumab administration to control leukocytosis but need to be discussed with the study PI * Previous treatment with immunotherapeutic agents (for example chimeric antigen receptor \[CAR\] T cells, long acting bispecific antibodies, etc) in the 28 days period prior to study drug administration on day 1 cycle 1, with the exception of short-half bispecific antibodies (blinatumomab) where the washout period is only 14 days * Requirement, at the time of study entry, for concurrent steroid \> 10 mg/day of oral prednisone or the equivalent, except steroid inhaler, nasal spray or ophthalmic solution * Use of immunosuppressant medications (other than steroid as noted above) in the 2 weeks prior to study drug administration (cycle 1 day 1) * Known central nervous system involvement. Patients with suspected CNS involvement must be evaluated by lumber puncture and be free of CNS disease prior to study entry. Previously treated CNS involvement is allowed provided adequate treatment has been provided and the patient is free of CNS disease * History of allergic reactions attributed to compounds of similar chemical or biologic composition to flotetuzumab * Any active untreated autoimmune disorders (with the exception of vitiligo) * Dementia or altered mental status that would preclude sufficient understanding to provide informed consent * Second primary malignancy that requires active therapy. Adjuvant hormonal therapy is allowed * Active uncontrolled infection * Significant pulmonary compromise * Unstable angina or clinically significant heart disease * Major trauma or surgery within 4 weeks before enrollment * Clinically significant uncontrolled illness * Known history of immunodeficiency virus (HIV) or hepatitis B or hepatitis C infection * Females only: Pregnant or breastfeeding * Any other condition that would, in the Investigator's judgment, contraindicate the patient's participation in the clinical study due to safety concerns with clinical study procedures * Prospective participants who, in the opinion of the investigator, may not be able to comply with all study procedures (including compliance issues related to feasibility/logistics)