Brain Tumor-Specific Immune Cells (IL13Ralpha2-CAR T Cells) for the Treatment of Leptomeningeal Glioblastoma, Ependymoma, or Medulloblastoma

Inclusion Criteria:

* Participant has verified leptomeningeal metastases
* Participant must have a Karnofsky performance status (KPS) \>= 60
* Participant must have a life expectancy of \>= 8 weeks
* If participant has a ventriculoperitoneal shunt, the valve must be programmable, and must be able to tolerate their shunts being turned off for 48 hours
* The effects of IL13Ralpha2-CAR T cells on a developing fetus are unknown. For this reason, women of child-bearing potential must have negative serum pregnancy test and agree to use a reliable form of birth control prior to study entry and for at least two months following study treatment. Male research participants must agree to use a reliable form of birth control and not donate sperm during the study and for at least two months following study treatment
* Participant has a histologically confirmed IL13Ralpha2+ tumor expression by immunohistochemistry (IHC) at the initial tumor presentation or recurrent disease (H-score \>= 50)
* Participant must have the ability to understand and the willingness to sign a written informed consent
* No known contraindications to leukapheresis, steroids, or tocilizumab

Exclusion Criteria:

* Research participant requires supplemental oxygen to keep saturation greater than 95% and the situation is not expected to resolve within 2 weeks
* Research participant requires dialysis
* Research participant has uncontrolled seizure activity and/or clinically evident progressive encephalopathy
* Failure of research participant to understand the basic elements of the protocol and/or the risks/benefits of participating in this phase 1 study. A legal guardian may substitute for the research participant
* Participant is unwilling to stop treatment with chemotherapy or endocrine therapy and/or radiation one week prior and during the first 4 cycles of the IL13Ralpha2-CAR T cell study
* Shunted participants either have a non-programmable shunt valve, or cannot tolerate their shunts being turned off for 48 hours
* Participant has a coagulopathy or bleeding disorder or cannot safely discontinue anticoagulation prior to placement of a Rickham reservoir
* Participant has a chronic or active viral infection of the central nervous system (CNS)
* Participant has any uncontrolled illness, including ongoing or active infection; participant has known active hepatitis B or C infection; participants with any signs or symptoms of active infection, positive blood cultures or radiological evidence of infections
* Participant is human immunodeficiency virus (HIV) seropositive based on testing performed within 4 weeks of signing the main informed consent
* Participant has an autoimmune disease
* Participant has another active malignancy
* Participant is unable to undergo a brain magnetic resonance imaging (MRI)
* Participant is pregnant or breast feeding. Because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with IL13Ralpha2-CAR T cells, breastfeeding should be discontinued if the mother wants to participate in this study
* 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)