Genetically Modified Cells (KIND T Cells) for the Treatment of HLA-A*0201-Positive Patients With H3.3K27M-Mutated Glioma

Inclusion Criteria: * Participants 2 to 25 years of age inclusive, at the time of signing the informed consent. The first two participants will be 12-25 years of age. * Male participants of impregnate potential must agree to use contraception, during the study and for at least 6 months after the last study intervention and refrain from donating sperm during this period. * Female participants of childbearing potential must agree to follow the contraceptive guidance, during the study and for at least 6 months after the last study intervention. * Females of childbearing potential must have a negative serum or urine pregnancy test within 14 days of receiving study interventions. * CNS reservoir such as Ommaya catheter must be in place. * Patients must be enrolled on PNOC COMP prior to enrollment on PNOC018 if PNOC COMP is open to accrual at the enrolling institution. * Participants with DMG who are positive for the H3.3K27M mutation (positive testing from a Clinical Laboratory Improvement Amendments (CLIA) laboratory required or equivalent) and who completed at least standard radiation therapy. * All participants must test positive for HLA-A\*0201 (positive testing from a CLIA or equivalent laboratory required). Other HLA-A2 subtypes are excluded. * All participants must consent for tumor tissue (fresh or archival) for biomarker analysis if available. * All participants must have evaluable or measurable disease at the time of consent. * All participants must be either off systemic steroids or be on a stable or declining dose of dexamethasone (maximum dose of 0.1 mg/kg/day or 4 mg/day) at the time of enrollment. * Participants must not have received any bone marrow transplants for the treatment of their tumor. * Participants must discontinue all anti-cancer agents and radiotherapy and, must have recovered from significant acute toxic effects of prior therapies. * Chemotherapy/biologic therapy: All cytotoxic chemotherapy/biologic therapy must be discontinued ≥ 7 days prior to enrollment. * Immunotherapy: The last dose of anti-tumor antibody therapy must be at least 3 half-lives or 30 days, whichever is shorter, from the time of enrollment. * All participants must have adequate organ function. * Adequate bone marrow function is defined as: * Peripheral absolute neutrophil account 1000/mm\^3 and * Platelet count 100,000/mm\^3 (transfusion dependent, defined as not receiving platelet transfusions for at least 7 days prior to enrollment) * Absolute lymphocyte count \>= 500/µL or CD3 count of \>= 150/µL * Adequate renal function is defined as: * Creatinine clearance or radioisotope glomerular filtration rate \>= 70 mL/min/1.73 m\^2 or * Maximum serum creatinine based on age/gender as follows: * 3 to \< 6 years =\< 0.8 mg/dL (male and female) * 6 to \< 10 years =\< 1.0 mg/dL (male and female) * 10 to \< 13 years =\< 1.2 mg/dL (male and female) * 13 to \< 16 years =\< 1.5 mg/dL (male) and 1.4 mg/dL (female) * \>= 16 years =\< 1.7 mg/dL (male) and 1.4 mg/dL (female) * Adequate liver function is defined as: * Bilirubin (sum of conjugated + unconjugated) =\< 1.5 x upper limit of normal (ULN) for age and * Alanine aminotransferase (ALT) and aspartate aminotransferase (AST) =\< 2.5 x ULN and * Serum albumin \>= 2 g/dL * Participants with a history of congestive heart failure at risk because of underlying cardiovascular disease or exposure to cardiotoxic drugs must have adequate cardiac function as clinically indicated. Only order ECHO and EKG for patients with history of cardiac toxicity. * (QTc) =\< 480 ms * Injection fraction \>= 45% by echocardiogram * Adequate pulmonary function is defined as no evidence of dyspnea at rest, no exercise intolerance due to pulmonary insufficiency, and pulse oximetry \> 92% while breathing room air * Adequate neurologic function is defined as a well-controlled seizure disorder and performance status (Lansky \< 16 years and Karnofsky \>= 16 years) that is at least 60. * Informed consent: Capable of giving signed informed consent or assent depending on participant age as appropriate which includes compliance with the requirements and restrictions listed in the informed consent form (ICF) and in this protocol. Assent will be obtained when appropriate based on the subjects age. Exclusion Criteria: Participants are excluded from the study if any of the following criteria apply: * Participants with MRI or clinical evidence of uncontrolled tumor mass effect; the assessment of mass effect should be made by the study investigators prior to any planned KIND T cell treatment. Pre-infusions MRI will need to be reviewed by the study investigators prior to dosing. Participants with an assessment score \>= 3 will be excluded, based on Table 4 in section 3.8.2 * Participants with a known disorder that affects their immune system such as HIV or hepatitis B or C, or an autoimmune disorder requiring systemic cytotoxic or immunosuppressive therapy. Participants who are currently using inhaled, intranasal, ocular, topical, or other non-oral or non-IV steroids are not necessarily excluded from the study. * Participants who have received prior solid organ or bone marrow transplantation. * Participants with uncontrolled infection. * Female participants of childbearing potential must not be pregnant or breast-feeding. * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements. * Untreated symptomatic hydrocephalus determined by treating physician. * Participants who are unable to return for follow-up visits or obtain follow-up studies required to assess toxicity to therapy. * Participants who are currently receiving another investigational drug. * Participants who are co-enrolled on another investigational protocol. * Participants who are currently receiving other anticancer agents (bevacizumab used to treat tumor mass effect will not constitute an exclusion; at time of enrollment participants need to be off bevacizumab and will need to be discussed with the study team).