The local cop is covering an incident, and takes offense when a higher authority shows up. The Fugitive, Die Hard, and a ton I’m blanking on. I’m gonna guess in real life, local guy would be relieved to have it taken off his hands, but in the movies he invariably takes great umbrage.
Dunno about movies but the standard vernacular here, when somebody tries to muscle in on somebody doing the task required is “Oi! Ned Kelly is robbing this coach”, or in extremis, “Who’s fucking this cat? You or me?”
Jurisdiction Friction seems like a good fit (and contains your examples). To me, it seems to be local cops vs. the Feds more often than not, and usually the FBI. Though whether the locals or the Feds are seen in a better light depends on the series.
No doubt it varies between jurisdictions. In the UK it’s usually the local “territorial” police vs. specialists like the Drugs Squad, or sometimes other agencies like Customs and Excise, or Immigration or even the “secret squirrels” of Special Branch or MI5. And woe betide our heroes if it should happen to involve the military. I’ve seen similar goings-on in continental police dramas.
But you can see how it would be perfectly possible for this or that villain to do something that connects completely different investigations, and for everyone to want to take priority on their turf.