When it comes to US flights, I believe if you set up a trip from the US to another country and the round trip back to the US, then both flights in this case are considered as doing business with the US and are required to follow their laws in terms of damages. Although I’m not sure if round-trip is explicitly required or if it just has to have a US stop somewhere.
I do wonder if an American can make a claim against an airline in the US even if the flight that caused the damage completely took place in another country, but the airline does regular business in the US too. Can they be sued? or is it out of courts jurisdiction? …hopefully I won’t have to ever find out.