Study Evaluating Brexucabtagene Autoleucel (KTE-X19) in Pediatric and Adolescent Participants With Relapsed/Refractory B-precursor Acute Lymphoblastic Leukemia or Relapsed/Refractory B-Cell Non-Hodgkin Lymphoma

Key Inclusion Criteria for the ALL Cohort * Relapsed or refractory B-precursor ALL defined as one of the following: * Primary refractory disease * Any relapse within 18 months after first diagnosis * Relapsed or refractory disease after 2 or more lines of systemic therapy * Relapsed or refractory disease after allogeneic transplant provided individual is at least 100 days from stem cell transplant at the time of enrollment * Disease burden defined as at least 1 of the following: * Morphological disease in the bone marrow (\> 5% blasts) * Minimal/Measurable Residual Disease (MRD) positive (threshold 10\^-4 by flow or Polymerase chain reaction (PCR)) * Individuals with Ph+ disease are eligible if they are intolerant to tyrosine kinase inhibitor (TKI) therapy, or if they have relapsed/refractory disease despite treatment with at least 2 different TKIs * Age ≤ 21 years and weight ≥ 6 kg at the time of assent or consent per Institutional Review Board (IRB) guidelines * Lansky (age \< 16 years at the time of assent/consent) or Karnofsky (age ≥ 16 years at the time of assent/consent) performance status ≥ 80 at screening * Adequate renal, hepatic, pulmonary and cardiac function defined as: * Creatinine clearance (as estimated by Cockcroft Gault or Schwartz) ≥ 60 mL/min * Serum alanine aminotransferase (ALT)/aspartate aminotransferase (AST) ≤ 5 x upper limit of normal (ULN) * Total bilirubin ≤ 1.5 x ULN, except in individuals with Gilbert's syndrome * Left ventricular shortening fraction (LVSF) ≥ 30% or left ventricular ejection fraction (LVEF) ≥ 50%, as determined by an echocardiogram or multi-gated acquisition scan (MUGA), no evidence of pericardial effusion (except trace or physiological) as determined by an echocardiogram (ECHO) and no clinically significant arrhythmias * No clinically significant pleural effusion, pericardial effusion or ascites * Baseline oxygen saturation \> 92% on room air Key Exclusion Criteria for the ALL Cohort * Diagnosis of Burkitt's leukemia/lymphoma according to the World Health Organization (WHO) classification or chronic myelogenous leukemia lymphoid blast crisis * History of malignancy other than non-melanoma skin cancer or carcinoma in situ (e.g. cervix, bladder, breast) unless disease free for at least 3 years * History of severe hypersensitivity reaction to aminoglycosides or any of the agents used in this study * Central nervous system (CNS) involvement and abnormalities: * Any CNS tumor mass by imaging and/or parameningeal mass (cranial and/or spinal) * Presence of central nervous system (CNS)-3 disease, defined as white blood cell (WBC) ≥ 5/µL in Cerebrospinal Fluid (CSF) with presence of lymphoblasts with or without neurologic symptoms * CNS-2 disease, defined as WBC \< 5/µL in CSF with presence of lymphoblasts and with neurologic symptoms (see note below for further clarification). * Note: Neurologic symptoms may include but are not limited to cranial nerve palsy (if not explained by extracranial tumor) and clinical cord compression. * (Individuals with CNS-1 (no detectable lymphoblasts in the CSF) and those with CNS-2 without clinically evident neurological changes are eligible to participate in the study) * History or presence of CNS disorder, such as cerebrovascular ischemia/hemorrhage, dementia, cerebellar disease, or any autoimmune disease with CNS involvement, posterior reversible encephalopathy syndrome (PRES), or cerebral edema with confirmed structural defects not related to lymphoma by appropriate imaging. History of stroke or transient ischemic attack within 12 months before enrollment. Individuals with seizure disorders requiring active anticonvulsive medication. * History of concomitant genetic syndrome such as Fanconi anemia, Kostmann syndrome, Shwachman-Diamond syndrome or any other known bone marrow failure syndrome * History of myocardial infarction, cardiac angioplasty or stenting, unstable angina, or other clinically significant cardiac disease within 12 months of enrollment * History of symptomatic deep vein thrombosis or pulmonary embolism within 6 months of enrollment. * Primary immunodeficiency * History of human immunodeficiency virus (HIV) infection or acute / chronic active hepatitis B or C infection. Individuals with a history of hepatitis infection must have cleared their infection as determined by standard serological and genetic testing per current Infectious Diseases Society of America guidelines or applicable country guidelines. * Presence or suspicion of fungal, bacterial, viral, or other infection that is uncontrolled or requiring IV antimicrobials for management. * Prior medication: * Prior CD19 directed therapy (other than blinatumomab), including CAR+ T cell, bispecific T cell engager (BiTE), and antibody drug conjugate (ADC), with the exception of individuals who received brexucabtagene autoleucel (KTE-X19) in this study and are eligible for re-treatment * Treatment with alemtuzumab within 6 months prior to leukapheresis, or treatment with clofarabine or cladribine within 3 months prior to leukapheresis * Donor lymphocyte infusion (DLI) within 28 days prior to enrollment * Any drug used for graft-versus-host disease (GVHD) within 4 weeks prior to enrollment * Acute GVHD grade II-IV by Glucksberg criteria or severity B-D by IBMTR index; acute or chronic GVHD requiring systemic treatment within 4 weeks prior to enrollment * Live vaccine ≤ 6 weeks prior to enrollment * Women of child-bearing potential who are pregnant or breastfeeding because of the potentially dangerous effects of the preparative chemotherapy on the fetus or infant. Females who have undergone surgical sterilization are not considered to be of childbearing potential * Individuals of both genders of child-bearing potential who are not willing to use a birth control method considered to be highly effective per protocol from the time of consent through 6 months after conditioning chemotherapy or brexucabtagene autoleucel (KTE-X19) infusion, whichever is longer. Key Inclusion Criteria for the NHL Cohort * Histologically confirmed aggressive B cell NHL * Relapsed or refractory histologically confirmed aggressive B-cell NHL per 1 or more of the following: * Primary refractory disease * Any relapse within 18 months after first diagnosis * Relapsed or refractory disease after 1 or more lines of systemic therapy * Relapsed or refractory disease after autologous /allogeneic stem cell transplant provided individual is at least 6 weeks from autologous stem cell transplant and at least 3 months from allogeneic stem cell transplant at the time of enrollment * Individuals must have received adequate prior therapy including at a minimum all of the following: * Anti-CD20 monoclonal antibody, unless the investigator determines that the tumor is CD20 negative * An anthracycline-containing chemotherapy regimen * Age \<18 years old and weight ≥ 6kg * Lansky (age \< 16 years at the time of assent/consent) or Karnofsky (age ≥ 16 years at the time of assent/consent) performance status ≥ 80 at screening * Adequate renal, hepatic, pulmonary, and cardiac function defined as the following: * Creatinine clearance (as estimated by Cockcroft Gault or Schwartz) ≥ 60 mL/min * Serum ALT/AST ≤ 5 ULN * Total bilirubin ≤1.5 x ULN except in individuals with Gilbert's syndrome * Left ventricular shortening fraction(LVSF) ≥ 30% or left ventricular ejection fraction (LVEF) ≥ 50%, as determined by ECHO or MUGA, no evidence of pericardial effusion (except trace or physiological) as determined by an ECHO, and no clinically significant arrhythmias * Baseline oxygen saturation \> 92% on room air Key Exclusion Criteria for the NHL Cohort * History of malignancy other than nonmelanoma skin cancer, carcinoma in situ (eg, cervix, breast), or follicular lymphoma (FL) unless disease free for at least 3 years * Autologous stem cell transplant within \<6 weeks of planned KTE-X19 infusion; allogeneic stem cell transplant within \<3 months of planned KTE-X19 infusion * Prior CD19 targeted therapy other than blinatumomab and loncastuximab tesirine-lpyl * History of severe, immediate hypersensitivity reaction attributed to aminoglycosides * Presence or suspicion of fungal, bacterial, viral, or other infection that is uncontrolled or requiring IV antimicrobials for management. * History of HIV infection or acute/chronic active hepatitis B or C infection. Individuals with a history of hepatitis infection must have cleared their infection as determined by standard serological and genetic testing per current Infectious Diseases Society of America guidelines or applicable country guidelines * Acute GVHD grade II-IV by Glucksberg criteria or severity B-D by International Bone Marrow Transplant Registry (IBMTR) index; acute or chronic GVHD requiring systemic treatment within 4 weeks prior to enrollment. * CNS involvement and abnormalities: * Any CNS tumor mass and/or parameningeal mass (cranial and/or spinal) by imaging with current or prior history of neurological symptoms within 3 months prior to screening. Note: CNS involvement without neurologic symptoms will be allowed. * History or presence of any CNS disorder such as a seizure disorder, cerebrovascular ischemia/hemorrhage, dementia, cerebellar disease, any autoimmune disease with CNS involvement, posterior reversible encephalopathy syndrome (PRES), or cerebral edema with confirmed structural defects by appropriate imaging. History of stroke or transient ischemic attack within 12 months before enrollment. Individuals with seizure disorders requiring active anti-convulsive medication. * History of myocardial infarction, cardiac angioplasty or stenting, unstable angina, or other clinically significant cardiac disease within 12 months of enrollment * Primary immunodeficiency * History of severe immediate hypersensitivity reaction to any of the agents used in this study * Live vaccine ≤ 6 weeks prior to planned start of lymphodepleting chemotherapy regimen * Individuals of both genders of child-bearing potential who are not willing to use a birth control considered to be highly effective per protocol from the time of consent through 12 months after the completion of lymphodepleting chemotherapy or brexucabtagene autoleucel (KTE-X19) infusion, whichever is longer. * Prior medication: * Prior CD19 directed therapy (other than blinatumomab), including CAR+ T cell, BiTE, and ADC, with the exception of individuals who received brexucabtagene autoleucel (KTE-X19) in this study and are eligible for re-treatment * Treatment with alemtuzumab within 6 months prior to leukapheresis, or treatment with clofarabine or cladribine within 3 months prior to leukapheresis * DLI within 28 days prior to enrollment * Any drug used for GVHD within 4 weeks prior to enrollment Note: Other protocol defined Inclusion/Exclusion criteria may apply.