Everyone Wants a Rescue Dog. Not Everyone Can Have One.

As a result, animal rescue has become a “retail operation,” said Greg Damianoff, director of the Bureau of Animal Regulation and Care, Houston’s municipal shelter. Rescue groups have been known to duke it out over the most desirable dogs like shoppers over a marked-down Armani gown. “You get a maltipoo in here and you’re likely to see a blood bath,” he said.

The hot commodities are the smaller, scruffy or fluffy dogs like the puppies Mr. Martinez was after. Shelters bundle loads of these dogs, known as the “cute and cuddlies” or easy-to-flip “flipper puppers,” with larger, older Labrador or pit bull mixes that are harder to move.

Mr. Damianoff sees sending off animals that locals may want as the only way to save the less desirable animals in his inventory. His shelter has a capacity of around 300 dogs and sometimes gets as many as 150 in a day. “We’re not trying to exclude anyone, but the whole thing is to get the animals out of here alive,” Mr. Damianoff said.

But as with many things that start with good intentions, there have been unsettling consequences, chief among them the recent deaths. Even when the animals arrive safely, they can carry parasites like heartworms and deadly diseases like parvovirus and distemper, which then spread not only in the receiving shelters but also to pets in the communities where the rescues are adopted.

Several states, including Maine, Massachusetts, New Hampshire and Rhode Island, have imposed more stringent entry requirements for rescue animals, like mandatory examinations by a local vet and quarantines. But these regulations are easy to circumvent. It’s not uncommon for rescue groups to take animals to neighboring states to make the handoff or to coordinate adoptions online, animal control officials say, making drops in parking lots in the dead of night.

In addition to relocating animals from less well-off areas of the mainland United States, rescue groups have begun bringing in dogs from China, Egypt, Mexico, South Korea, Thailand and Puerto Rico. United States Customs and Border Protection and the Centers for Disease Control and Prevention have raised security and public health concerns about this trend. In May, the C.D.C. suspended all imports of dogs from Egypt after several rescue dogs arrived with rabies.