It happened to me three times. Since one of the incidents occurred on the last leg at home base I just went home. However on the other two incidents we simply got a new airplane and continued on our way.
It is a different story for our F/A's as their contract states that they have the right to return to home base. In my case’s the F/A's decided to stay with the operation.
I should add on the last engine failure I did get a call from the CP on the bridge just before departure asking me whether I was OK to continue. I suppose he had to do that to protect his behind but it is rather novel.
Can you visualize how the passengers would feel if the original pilots and F/A’s booked off and they the passengers, didn’t?