No other judicial authority has jurisdiction? Are you serious?
You are aware of how juresdictions operate, even slightly. Right?
In general, it goes form Federal, to State, to Local. Each one having more authority over the lower one. That is why somebody awaiting trial (or punishment) on state or local charges may see the Prosecutors told to "pound sand" as Federal authorities take over the case and take priority in prosecution. And why a prisoner in Local or State custody can be snatched out of that by Federal Marshals or the FBI. Once again, Federal trumps State.
On board a military facility, only the Federal Government has authority. Even if it is a civilian attacking a civilian, it is almost unheard of for that case to be sent to a local or state court. Instead, because it happens on a military base, it is automatically a Federal crime. Civilian law enforcement has absolutely no authority there, unless they themselves decide to push it to a lower court for specific reasons.
Say if somebody was facing Federal charges for say fraud, but also State charges for murder. Then in a case like that the Federal court may indeed decide to not hold the trial, and let the state handle it. But this is still at their own discretion.
But a military member on a military instillation? Nope, no other court in the land has the right to take that case away. Not even a Federal Court. Unless once again in the case where tthe charges there were of a more serious note, and the military decides to give them up and handle their own prosecution at a later date.
Hell, go back to a common trope in movies and TV shows for decades, it is actually largely true. Being chased by cops in one jurisdiction, flee to another and they can largely just stop at the border and do nothing. Some do indeed are given cross-jurisdictional authority, but those are given between them previously. Cross one to enter another even in a "hot pursuit", and the cops themselves can even be arrested.
This has actually happened many times over the decades when some cop decides they can follow somebody onto a military base. Which has even seen them arrested themselves by base security as they have no legal right to be there. I myself have had gate guard duty, and nobody enters without authority. Not even other federal law enforcement agencies. Bases are autonomous unto themselves. We have even held INS agents at the gate and refused to let them in, as we watched dozens of illegals flee our base. They had no legal authority to enter, and we made sure they knew that.