“If he throws up or gets sick or goes bonkers, there’s going to be a human being there,” Mr. Rueff said. “That makes it worth it for us; we’re paying for peace of mind.”

Airlines have always had varying restrictions on animal travel. There are a few carriers, like Alaska Airlines, Frontier Airlines, Hawaiian Airlines and Sun Country, that still allow brachycephalic breeds to fly in cargo. And most animals are generally allowed to fly in the passenger cabin if they weigh less than 20 pounds, as some French bulldogs and many pugs do.

But the clear trend among commercial airlines is toward an outright ban on brachycephalic breeds.

American Airlines banned brachycephalic breeds of dogs and cats shortly after four bulldogs died on its planes in a three-month period in 2010. Delta stopped accepting French, English and American bulldogs this year, after three bulldogs died from January to March.

United and Continental Airlines, which had two bulldogs die in their care this spring, banned brachycephalic dog breeds from flying during the summer, lifting the restriction on Sept. 16, when temperatures began to drop.

According to the federal Agriculture Department, 189 animals died on commercial flights from June 2005 to June 2011; of those animals, 98 — more than half — were brachycephalic breeds.

The breeds, which also include Persian and Himalayan cats, have smaller openings to their noses and elongated soft palates on the roofs of their mouths, which make breathing more difficult for them, veterinarians said. Those breathing problems can be magnified in stressful situations like air travel, and further exacerbated in extreme heat.