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“This is helping people keep their pets,” said Dawn Lauer, the Humane Society’s outreach coordinator. Ms. Lauer said the organization has given away $113,000 in shelter grants, which have been used for pet food banks, health care, foster programs, even paying pet deposits for people who are seeking rental homes.

Mr. Zawistowski said that thinking ahead can prevent a last-resort situation for homeowners who are forced to move. Usually, one can see a foreclosure coming. If that’s the case, he said, talk to friends and family familiar with your pets as early as possible. If that doesn’t result in a home, talk to your veterinarian and rescue groups, he said.

“They may be able to help provide foster care — while you look for housing that will take your pet,” he said. “It is helpful if they are kept up to date on vaccinations. This is also where obedience skills will make for better houseguests.” Mr. Zawistowski said Washington has an advantage compared with other areas of the country because it has a strong network of animal-rescue groups to help families in need. Those groups are getting more calls, he said.

“Rescue groups are acting as a buffer and absorbing some of the pets before they hit the main shelters,” Mr. Zawistowski said. “At some point, there will be a spillover.”