Please don't give out the impression it's easy as it isn't and don't forget all countries have different rules and regulations along with the airlines. For example a dog coming from the UK doesn't need a microchip but from France it does.
Also it is not as simple as picking up your dog from AFCD. You need the AWB and go to the correct cargo terminal (there are 3 main ones).
Do your research and don't take heed to advice from a blogger in the US. Goto official government sites for the up to date information.
A far more useful site then the one above is pettravel.com You can search country regulations and airlines. Remember all airlines have their own policy on how they transport pets. For example UA won't take crates over PP90 size or accept any snubs. AA will not accept pets between US and HK. BA for several weeks this year had an embargo on all dogs but accepted cats.
Things are changing all the time so please research official sites and not ones where someone did it a few years ago.