AA are particularly bad about this in my experience. When I was traveling with my spouse (pregnant at the time), we misconnected LHR-ORD-EWR because ORD-EWR was cancelled because of high winds (if I remember correctly). Even so, we couldn't have made the scheduled EWR flight anyway given the time it took us to get through immigration and customs at ORD that day.

Getting AA to put us up in a hotel was like squeezing water from a stone. I finally said, "Do you think it's reasonable that a pregnant woman booked in a first class transatlantic ticket should sleep on the floor of the airport because you cancelled a flight?" They okayed a hotel and thanks for Flyertalk I made sure we got a voucher for the one connected to the terminal so no bus needed.

The whole problem would have been avoided if there had been award space LHR-JFK but there wasn't any. Generously, the checkin at LHR offered to switch us to the nonstop to JFK for free but it was too late for them to be able to do it with checked baggage given the departure times.