Well here’s one pitched right down the middle at me. I’ve flown private jets for some years now, although I’m probably transitioning out of the industry.
It depends. Some airports that are typical international destinations for private jets (Teterboro, for example) have dedicated customs facilities. After landing we taxi there, shut down engines and wait for an officer to come out to us. They then process the passengers and crew there, or sometimes have us go inside. Then we start up again and taxi to the jet terminal (called an FBO, short for fixed base operator). Occasionally, the FBO will have sent over a shuttle bus for the passengers, but we still have to move the airplane away from the customs building.
At airports without that facility a couple of things can happen. We can get customs to send agents over to our airport of intended landing. That requires advance notice and the hours and availability can vary. I once had to divert from JFK to Farmindgale on Long Island at the last moment because of weather. My company had to hustle to get customs over to meet us, which was not a sure thing. It worked out, but could have been a major nuisance. In other cases we may be forced to land at another airport to clear customs, then continue to the final destination.
It’s mostly the same in other countries, except some places are super strict. I never know whether we are allowed to open the aircraft door after shutdown because some places go absolutely ape if you do. So we keep it closed, which often results in an officer knocking on the door asking why we haven’t opened it. Which is to say, it’s not really standardized.
One time I flew a guy from Bermuda to New Orleans and they wanted us to go inside to meet customs as if we were a commercial flight. They had us block into an airline gate, which I found funny and thrilled my partner because he had never been an airline pilot. That was weird.
But in the end, passengers and crew are subject to the same process and customs agents definitely to not have a sense of humor about lapses. I know of a crew that took people from NY to Greece, not knowing one passenger forgot their passport (usually we check before departure). They got there, the guy was detained in an office while the crew got their required rest, then they had to take him right back to NY.
Also knew of a crew that just plain forgot to go to customs after landing. They dropped off their pax at the FBO as usual and went to the hotel. Customs was not amused and all sorts of hell was raised.