If she took the job, she and her husband decided, they would not uproot their lives. Instead, they would rent an apartment for Ms. Rizzo, and Mr. Pavich would visit as often as possible.

“All our friends and family are in Vancouver, and it is a pretty hot market,” Ms. Rizzo said. “If you sell and try to come back, you can never re-enter the market.”

Mr. Lieberman was prepared with a lineup of buildings. A few were in Dumbo, including 220 Water Street, a converted shoe factory. A lofty, sunny one-bedroom there with almost 700 square feet had plenty of character. The rent was $3,825. But the location wasn’t sufficiently Nets-friendly, with a long, drab walk to the arena that would lead her under a highway overpass.

Image A one-bedroom unit at 220 Water Street, a converted 19th-century shoe factory, had plenty of character. But it would have made for an inconvenient commute. Credit... Robert Wright for The New York Times

For $4,000, a charming two-bedroom in a Brooklyn Heights brownstone occupied an entire floor, with around 1,200 square feet. Such a building, run by a small-property owner, can work well for international clients, Mr. Lieberman said: “The owner is more willing to have flexibility if the tenant doesn’t have U.S. credit.”

Inside, however, it seemed more quirky than charming, with an uninviting kitchen, and it couldn’t compete with the area’s gleaming, amenity-filled new buildings.

The two-year-old rental tower at 300 Ashland Place, with almost 400 units, was ideal: steps from both Barclays Center and Atlantic Terminal. From there, it was one express stop or an easy bicycle ride to Industry City.