Jurisdiction over the local cops, the county Sheriff, the Federal Marshalls and the FBI. By my observation of photos, etc, published here at FR, the Texas Rangers seemed to be conspicuous by their absence, in kinda the same way they excused themselves from the scene at the Branch Davidian / David Koresh compound near Waco when the ATF arrived on the scene what, about 30 years ago, which is to their credit.
Let's hope that Governor Abbott follows through on his earlier statement about putting the Rangers in charge of the investigation.
Here’s the law
https://codes.findlaw.com/tx/education-code/educ-sect-37-081.html
exas Education Code - EDUC ยง 37.081. School District Peace Officers, School Resource Officers, and Security Personnel
That does not mean that "Chief" Arredondo could not have handed operational control over to the Uvalde PD, the Uvalde County Sheriff's Office, or the Texas Highway Patrol. Rural PDs and SOs routinely, and gratefully, hand murder investigations in their bailiwick over to the Texas Rangers. Arredondo did not even have to stop being the incident commander -- he could have appointed an operations commander to handle tactical planning and execution. That is straight out of the National Incident Management System (NIMS), and I guarantee you "Chief" Arredondo has papers, or PDFs, in his personnel file indicating that he has had extensive NIMS training.