If you use a car seat at home, why would you not want to use one abroad? Would unfamiliar roads, new traffic rules and driving on the “wrong” side of the road make you feel more or less secure about your child's safety?
That said, we'd generally like to avoid the hassle of carting along a car seat if we could avoid that by renting one with the rental car. Our experience has been that you cannot reasonably expect to rent a car seat in most parts of Asia; they are just not used as much as in the U.S.
Since driving a rented car through Asian paradises like urban Seoul, downtown Jakarta and other race tracks is somewhat menacing anyway, the lack of rental seats might be a clue as to why renting a car may be a bad idea of its own.
Europe is a better picture, in that the need for car seats is better seen, though renting one is spotty.
One kind reader in the UK wrote us to say “In my experience of car-hire in Europe, child seats are few and far between. You have to book well in advance for a baby seat and booster seats are virtually nonexistent in Spain, Italy or Greece” (Thank you Debbie!)