United officially stopped flying brachycephalic breeds (more commonly known as snub-nosed dogs or cats) last June, as the dogs’ compressed facial structure can compromise their ability to breathe. United also no longer accepts dogs that require crates taller than 30 inches. In March, Delta Air Lines stopped transporting large dogs requiring crates taller than 24 inches, which effectively grounds large breeds like Labrador retrievers. American Airlines still ships large dogs, but much of their fleet cannot actually handle the larger crates, says Jeni Redmond of petsfly.com, a Phoenix-based pet relocator.

Mr. Lim turned to Globy, a Beijing-based pet relocation company, to help him book a flight through Lufthansa, one of the few international airlines that still accept snub-nosed pets in the cargo hold. This meant Mr. Lim flew alone to Washington while his spouse accompanied Bua Loy flying through Europe, with a lengthy layover in Frankfurt. The transport fee for the dog was $450 and a 12-hour journey was stretched to 42 hours.

“The last couple of years has seen all of the airlines … putting more restrictions on pets flown as cargo to alleviate their risk,” Ms. Redmond said. When the case of a pet tragically dying during air transport makes headlines, as recently happened when a husky died in the cargo hold of an Air France-KLM flight, airlines are tempted to further restrict animal travel.

The recent and not-so-recent restrictions are confusing, as there are different rules for different airlines. Traveling with your pet as carry-on or checked baggage is usually the most convenient option, and it is relatively inexpensive, with carry-on generally starting at about $100 for a pet on a domestic flight. And breed bans typically do not apply to pets that can be carried onto the cabin and fit inside a soft carrier that slides under the seat ahead.

Animals can also be sent alone as air cargo, and air cargo usually (but not always) accepts a broader range of animal breeds and sizes. This is more expensive, and can run into the tens of thousands for particularly large breeds.