As Democrats and Republicans in Washington negotiated legislation to overhaul the asylum system and tighten border security earlier this year, many in Cochise County, Ariz., a conservative stronghold, held their breath.

The bill promised to secure more of the federal grant funding that had been helping local mayors and community leaders house and bus migrants who were passing through their stretch of the U.S.-Mexico border. But after former President Donald J. Trump helped sink the proposal, even some of President Biden’s staunchest Republican critics in the area said that they were disappointed.

And yet, as news spread this week of Mr. Biden’s latest action, which seeks to follow through on some of the asylum provisions of that failed legislation, Republicans and Democrats alike in the area, a place where many have long felt their needs are ignored, eyed it with skepticism.

Mayor Donald Huish of Douglas, an even-keeled Republican, wondered whether some action was better than none.

“On one hand, I am happy to get any help we can get,” he said. “On the other, I wish they would just fix the problem.”