YORK, Pa. — For Republican Rep. Scott Perry, the influx of Central American children who have crossed illegally into the United States has propelled immigration to a top concern for voters in his heavily rural district far from the Texas border, eclipsing the new health care law and the federal deficit.

“I think people are very upset, and people have really been awakened to the immigration issue where they haven’t been before,” the first-term congressman from southeast Pennsylvania said in an interview at a bus leasing company, where he recently met with a group of small-business owners. “Right now at this current time, I would say immigration is the No. 1 issue on people’s minds.”

It’s the same story around the country this summer as polls show the crisis of unaccompanied children at the border has made immigration a pivotal issue with November elections approaching.

Republican Senate candidates in three contested races have focused ads on the issue, and it has the potential to affect campaigns in unpredictable ways that hold risks for members of both parties and for President Barack Obama.

For now, Republicans like Perry are able to boast that the House took action to address the border crisis before leaving Washington for its August recess, even though the Senate and Obama did not.

Republicans “demanded that we stay and pass a bill so we could show the American people ‘This is what we stand for,’” Perry told the business owners, referring to the House GOP’s legislation to spend $694 million on the border and make controversial policy changes to return the migrants home more quickly, as well as end an Obama program that granted work permits to more than a half-million immigrants brought to the U.S. illegally as kids.

How the issue plays out over the fall depends both on what happens in South Texas, where border arrivals have declined in the summer heat, but could rise again, and in Washington, where Obama is weighing extending deportation protections and work permits to millions more people already living in the U.S. illegally.

Such a move could upend the politics around immigration yet again, thrilling Latino voters who will be crucial for the 2016 presidential election. But it could also rile up Republican base voters, who are more likely to turn out this November and could make the difference in a handful of GOP-leaning states where vulnerable Democratic incumbents are trying to hold on.

Perry and other Republicans warn the president would pay a steep price politically for taking such a step.

“I think there will be a backlash, not necessarily that people will automatically come to vote for Republicans, but like in so many elections they might just stay home because they’re disgusted,” Perry said.

Indeed, Senate Democrats seeking re-election in red states, including Arkansas’ Mark Pryor and North Carolina’s Kay Hagan, have cautioned Obama against proceeding unilaterally.

But there’s also a risk for Republican lawmakers such as Perry and Rep. Joe Pitts, whose district borders Perry’s and includes Lancaster. They already are hearing from angry constituents who want Obama impeached, and executive action by the president would likely only increase such demands. That’s an unwelcome prospect for most Republican officeholders who see impeachment as a political loser, since it would be certain to energize Democratic voters and likely turn off many mainstream Republicans.

“It’s just absolutely ridiculous. We’re not going to do that,” Pitts said in an interview in Quarryville, 40 miles east of York through rolling green hills, after a local dairy farmer declared that any president should be “automatically impeached” for taking as many executive actions as Obama has.

The chairman of the Democratic Congressional Campaign Committee, Rep. Steve Israel, D-N.Y., said Republicans could end up in trouble if Obama’s moves on immigration increase calls for impeachment.

“The problem that Republicans have right now is that they have engineered a strategy to turn out their base voters in a midterm election and that may backfire against them as their base voters demand that House Republicans keep going farther and farther to the right,” Israel said in an interview.

Rep. Greg Walden, R-Ore., who leads the National Republican Congressional Committee, countered in a statement that voters want someone who will secure the border, and “Republicans have a very clear and consistent message about the need to provide an appropriate check and balance on this administration.”

Still, the border crisis has already scrambled the politics of immigration. Establishment Republicans have feared that given the growing number of Latino voters, they would pay a political price over their inaction on comprehensive immigration legislation, which died this year in the House. That may prove true in the presidential election in 2016, but so far this year Democrats have sometimes been on the defensive, as polls show the southern border crisis has caused support for comprehensive reform to dip while voters embrace calls for border security.

“Want to know why there’s lawlessness on our border? Ask Sen. Shaheen,” Republican Senate candidate Scott Brown asks in one ad against incumbent Democrat Jeanne Shaheen of New Hampshire. “She voted against border security twice, and for amnesty. It’s time for us to secure the border and enforce the law.”