In the Army, if there was a question of jurisdiction, we used to separate local cases by severity of penalty and throw the offenders to the authorities most likely to produce the severist sentence. This meant all military-based infractions went to us and crimes carrying hard time under local state or federal law, but not so much under UCMJ, - like drug offenses - went to the local civil authorities. We always worked with local cops, state troopers or patrol, local FBI, on post MPs and the 902nd (national security crimes). Whoever had the most logical jurisdiction was welcome to pursue the investigation, as long as everyone was in the loop.
In homicides, if it was on post involving military persons, the on post Criminal Investigation team usually took the lead. If help was needed, they could request local assistance. The FBI was usually called in on National Security crimes & worked closely with the CounterIntelligence agents involved.