“What people are seeing now is that the demographic trends in North Carolina and several other Southern states are moving in an unchangeable direction toward more progressive politics,” Mr. Foxx, a Democrat, said.

Bolstered by carefully drawn legislative maps and the results of nonpresidential elections, in which minority turnout usually dips, Republicans have maintained an iron grip on state government in places like Georgia and North Carolina, as well as some Western states.

But Republican leaders acknowledged that a moment of reckoning was ahead, perhaps sooner than expected because of this year’s presidential contest. State Representative B. J. Pak of Georgia, a Republican of Korean descent, said that he doubted that Mrs. Clinton would win the state but that his party’s leaders were acutely aware of the fast-changing terrain.

Mr. Pak, who represents Gwinnett County, a suburban area near Atlanta where minorities now outnumber whites, said it had been hard this year to recruit a diverse slate of candidates, or to win over women and minorities.

“Donald Trump has really made it extremely difficult,” said Mr. Pak, who has not endorsed Mr. Trump. “They feel that the party’s not welcoming, and that’s a tremendous challenge when you’re trying to get people to give the party a chance.”

Bob McDonnell, the former governor of Virginia, whose 2009 election marked the last major Republican victory in a once-reliably red state, said his party must embrace “the new Americans, the immigrants that are really kind of remaking America, like how the Irish did in the mid-1800s.” He noted that he had learned phrases in Korean and Mandarin to campaign in Northern Virginia.

“We get accused, as conservatives, of writing off certain groups of people because they don’t fit a certain political stereotype. You can’t do that as a Republican,” Mr. McDonnell said. “It’s a recipe for certain minority status.”