...to the extent that the airline knows what your real destination is.

If you are connecting on a single ticket, they know, and they're responsible for getting you to where you want to go.

If you have two separate tickets, the first airline got you to your destination (as far as it's concerned) when you arrived at the connecting airport. The second airline doesn't know or care how you get to their flight, as long as you get there before the doors close. If you don't, it's your problem. Their responsibility to take care of you in that situation depends on the conditions of your ticket.

This is a reason to book an entire trip on a single ticket whenever possible. (It sometimes isn't possible, as when one segment is an award flight and the next is paid.)