Is it legal/possible to request a flight-path change from Domestic to International minutes after take-off, for a privately owned jet? This is related to the recent escape of Carlos Ghosn from Japan. While we don't know the details yet, M. Ghosn allegedly boarded a private plane from a small airport near Osaka [and probably in a kind of "luggage"!].

In this event, it seems the private plane was aimed to travel internationally. But that brings the question anyway:

Would the pilot of a private plane be authorized to change their destination after take-off, and, more specifically, request to travel to a foreign destination, while initially, (before take-off) the plane was supposed to reach a domestic destination?

Probably not the case in this escape story, but since there is much less passengers verification on domestic flights, allowing the flight-path change (for a private plane) would make things much easier for someone (wealthy) who wants to evade a country.
Generally speaking, ATC doesn't care who you are or where you're going. Their only job is "a safe, orderly and expeditious flow of traffic." If you request to amend the destination on your flight plan, they'll just type it into their computer and give back the new clearance it spits out. Diverting is a fairly routine thing, actually, especially when bad weather is afoot. If they even ask why, it's to make sure it's not due to an emergency or urgency. What (if anything) happens once you're safely on the ground at that new destination is someone else's problem.

That said, the militaries of many countries may take exception to someone flying into their airspace without prior permission, which is a much more complicated topic.