Phase I Trial Integrating HLA-Haploidentical Anti-CD19 CAR-T Cells With Post-Transplantation Cyclophosphamide-Based HLA-Haploidentical Hematopoietic Cell Transplantation

-INCLUSION CRITERIA - Recipient

1. Participants with high or very high-risk hematologic malignancies, as defined by the revised Disease Risk Index (DRI), or malignancy that remains persistently MRD+ (by flow cytometry, cytogenetics, FISH, PCR, or NGS) on most recently assessed disease specimen (within 2 months of initiating conditioning).
2. Hematologic malignancy must be CD19+ (uniform expression on immunohistochemistry or \>= 80% on flow cytometry) as confirmed by CD19 IHC assay (BT51E) or flow cytometry (BD QuantiBRITE(TM) Beads PE Fluorescence Quantitation Kit). (Participants do not have to have refused or lack access to commercial anti-CD19 CAR-T-cell therapies since this study focuses on the integration of CAR-T cells and HCT and not specifically the CAR-T cells themselves; furthermore the construct used to manufacture this product is the same as used in a current commercial product, but developed from a new batch and with a similar but not identical manufacturing process.)
3. Age 18-75
4. Karnofsky \>= 60%.
5. Participants must have adequate organ and marrow function as defined below:

   * Cardiac ejection fraction \>= 45% by 2D echocardiography;
   * Forced expiratory volume-1 (FEV-1) and diffusing capacity of the lung for carbon monoxide (DLCO) (corrected for hemoglobin) all of \>=50% predicted (this requirement would be waived in participants who are unable to properly perform pulmonary function tests - in such circumstances, participants must have pulse oximetry \>=90% on room air and no dyspnea or obvious pulmonary restrictions);
   * Estimated serum creatinine clearance of \>= 60 ml/minute/1.73m\^2 calculated using eGFR in the clinical lab (participants with estimated serum creatinine clearance less than 60 may have measured creatinine clearance performed and if \>= 60 will be considered eligible);
   * Total bilirubin \<= 2X the upper limit of normal (participants with documented or suspected Gilbert s are exempt from this requirement);
   * Alanine aminotransferase (ALT) and aspartate aminotransferase (AST) \<= 5X the upper limit of normal.
6. At least one available HLA-haploidentical donor
7. Women of child-bearing potential (WOCBP) must agree to use a highly effective method of contraception (hormonal, intrauterine device (IUD), surgical sterilization, abstinence) at the study entry and for 1 year after transplant (restriction period).

   Men must agree to use an effective method of contraception (barrier, surgical sterilization, abstinence) at the study entry and for 1 year after transplant. We also will recommend men with female partners of childbearing potential to ask female partners to be on highly effective birth control (hormonal, intrauterine device (IUD), surgical sterilization). Men must not freeze or donate sperm within the same period.
8. Breastfeeding participants must be willing to discontinue breastfeeding from study treatment initiation through 1 year after transplant.
9. Participants seropositive for human immunodeficiency virus (HIV) not due to intravenous immunoglobulin, must have adequate viral suppression (HIV viral load \< 200 viral copies per ml of blood) prior to the beginning of conditioning.
10. For participants seropositive for hepatitis B virus (HBV) core antibody not due to intravenous immunoglobulin, a HBV viral load should be undetectable.
11. Participants seropositive for hepatitis C virus (HCV) not due to intravenous immunoglobulin must have been treated and cured. For participants with HCV infection who are currently on treatment, they are eligible if they have an undetectable HCV viral load.
12. Ability of participant or legally authorized representative (LAR) to understand and the willingness to sign a written informed consent document.
13. Ability and willingness of participant to co-enroll on 20-C-0051: Gene Therapy Follow Up Protocol for Subjects Previously Enrolled in NCI for Immuno-Oncology Studies

14 Willingness to remain in the NIH hospital or, if discharged, stay close to the NIH (\<60 minutes drive), for a minimum of 100 days after transplant or longer if there are complications. Participants must commit to having an adult caregiver with them during the first 100 days after transplant in case of discharging from the hospital before 100 days.

INCLUSION CRITERIA - Donor

1\. Related donor (age \>=12) deemed suitable, eligible, and willing to donate, per clinical evaluations, who are additionally willing to donate blood, bone marrow, and stool for research. Related donors will be evaluated in accordance with existing Standard Policies and Procedures for determination of eligibility and suitability for clinical donation.

EXCLUSION CRITERIA - Recipient

1. Participants who are receiving any other investigational agents within 3 weeks prior to the beginning of conditioning.
2. Active CNS involvement of primary hematologic malignancy
3. Active malignancy of non-hematopoietic type (excluding non-melanoma skin cancers) which is metastatic, relapsed/refractory to treatment, or locally advanced and not amenable to intended curative treatment per standard of care.
4. Prior checkpoint inhibitor therapy within 6 weeks prior to the beginning of conditioning.
5. Prior history of seizure.
6. Uncontrolled infection.
7. History of allergic reactions attributed to compounds of similar chemical or biologic composition to study agents used in study.
8. Positive beta-HCG serum or urine pregnancy test performed in females of childbearing potential at screening. (A low positive test in a post-menopausal woman may not be exclusionary if deemed not indicative of pregnancy per gynecology.)
9. Uncontrolled intercurrent illness evaluated by medical history, physical exam, EKG, and laboratory testing (e.g., severe endocrinopathy, disseminated intravascular coagulation, profound electrolyte disturbance) that would make it unsafe to proceed with transplantation.

EXCLUSION CRITERIA - donor

1\. Pregnancy