But they are a familiar part of the landscape. They are Christians. They have been a big part of the community for decades. Muslims, however, are none of those things. Most of what people know comes in the form of the daily drip of news into their iPads, and that does not leave a good impression. So the part of the immigration message that really resonates here is about Muslims from the Middle East.

“It’s a little different than any time in history, when we are seeing this level of terrorism sweeping the world, and that has gotten people’s attention,” said Bill Campbell, pastor at Hendersonville Presbyterian Church. “Terrorism is an ideology, it’s not a religion, but the religion that tends to give birth to it most often these days is Islam. The threat from that is very real, and we can’t just ignore it.”

And while Mr. Trump has bounced around on the topic since Election Day — on Wednesday his spokesman indicated that suspending immigration from countries with a lot of terror attacks was still part of his plan — many moderate voters say they are hopeful about where he will eventually land.

“I’m telling you, he’s going to surprise people, he’s going to make people mad on both sides,” said Greg Mathis, senior pastor at the Mud Creek Baptist Church in Hendersonville. “Sometimes he’s too blunt, maybe a little too raw. But securing our borders has been a long time coming, and Donald Trump was the first one in a long time to say it. I think that’s what registered with so many people in this country.”

American Muslim organizations say that singling out Muslims amounts to racial profiling, and that Mr. Trump has deliberately stirred the pot, promoting theories about Muslims that simply are not true. They say that violence comes from ideological zealots, not from Islam. They say that the United States already does extremely vigorous vetting of immigrants and that some recent attacks that cited the Islamic State were carried out by Muslims who were born in the United States.