Well...I just got done watching "Jersey Devil". Mulder (with Scully mostly along for the ride) investigates a murder on his own voiliton. And yes, he has no authority, he's not even there to take over. Hard-headed local cops, that's fine. But throwing an FBI agent in jail for three days on trumped up drunk and disorderly charges? C'mon.
It's understandable when Kolchak is the bottom of the authority food chain. (Hence my comment about them 'having as much authority as a Chicago newspaperman'.) but you're FBI SPECIAL AGENTS. Throw a little weight around. No you can't take over without authority. But, you are not Barney Fife either.
Disclaimer:I distinctly remember this happening a lot in the series. But hopefully I'm conflating this happening with 'mythology episodes', where it's perfectly natural for them to be tossed around.