You can't tell from their accents—the dogs aren't big on conversation—but anyone getting acquainted with the rescue hounds at North Shore Animal League's adoption center will learn something surprising: Most of these pooches aren't from around here. Rocky, a Basset mix, is from North Carolina; Zeus, a handsome Collie, hails from Tennessee; and Cathy, a perky young Beagle, just drove up from Virginia.

It's the same scene at any dog run in New York City. Ask the owners, and you'll hear it over and over again: "I rescued...