With the recent switch to all electronic everything that seems to be taking place one might want to take the time to locate the appropriate documents for operation of the flight and aircraft. The class I had received on aviation authority inspections in a foreign country was invaluable on this topic.
Anecdotally, this can indeed be a show stopper if the Federales somewhere need to see some insurance or training certificate that is now stored 'in the cloud'. Some authorities still only recognize paper documents and finding someone in ops several timezones away who knows how to send an international fax can be a challenge these days.
And, I've heard of a European authority detaining an aircraft until someone could find their FCC Radiotelephone Operator Permit. I thought those had gone the way of the ICAO crewmember certificate that we used to carry. But, I'm sure I have my FCC permit here somewhere, let me look...
As far as I can remember, I've only been checked on the line by the feds once over the years. I have had the FAA (and a couple of DGCA's) observe training in the sim and the plane as is traditional.
I was an FO in WAW doing the walkaround for an ETOPS crossing when the 'friendly stranger' approached and showed me his credentials. He was from the FAA. Turns out he was in Poland to work with U.S. certification of some agricultural aircraft, probably an AN-2 variant. He and his mechanic partner actually had tickets but he wanted to ride up front so he gave us an air carrier inspection which was professional but also very pleasant.
When it was time to start down, the fed asked if the FAA mechanic could ride in the cockpit for the landing. These days that would probably be a trick question involving regs, tables, the TSA and the FCOM. Anyway, the captain said OK and the only debrief was warm thanks for the ride.