International warranty for Mac book Air

Yes. The warranty on non-cellular devices is world-wide. Note that if you need things like a keyboard replaced, there may be significant delays or you may be referred to the country where it was originally sold as Apple Stores don’t typically stock things like replacement keyboards for other regions.

Keep in mind with many of Apple's laptops, the battery is considered part of the Top Case Assembly which includes the Keyboard so replacing the battery would incur the same delays or being referred to the country of origin for repairs. Apple does allow the replacement of just the battery in some models such as some MacBook Airs (even the M-series), but AFAIK none of the MBPros have that option at this time.