I operated a flight where having been airborne for a while we found a passenger who was trying to get to a different destination with a different airline. Tracing back in the later wash-up, just as he presented his ticket at the gate, which was next to the one he should have used, the agent collapsed and in the ensuing commotion the passenger carried on to the aeroplane. By another coincidence, the same seat number was free (out of only a handful of spare seats) and what seemed like his name on the ticket was actually his original destination.
A very diligent crew member picked up that something was not quite right during the flight and examined his ticket, finding out he was meant to be about 2,000 miles away. His English was very limited and he appeared blissfully unaware of the error (and enjoying his Premium seat) so we arranged for a connecting flight to take him back to where he should have landed when we got to our final destination. To this day I think he never worked it out, although the flight was four hours longer than it should have been...
what would be the problem with simply carrying on with the flight?
I wondered about that too - it could be a directive from a higher power as one would assume that the crew had asked for guidance in the matter?