David Kim, a Korean-American Democrat running for Congress in the suburban Seventh District, won enough votes on Tuesday to earn a spot in a runoff. Sheikh Rahman, a Bangladeshi immigrant, effectively won a State Senate seat on Tuesday by winning a Democratic primary, because no Republican is running for the seat.

Mr. Kim said in an interview Wednesday that he had reached out to black, white, Latino and Asian voters in his district. “The Seventh is, from a diversity point of view, a melting pot of America, and what I’d call the New South,” he said.

As the suburbs have evolved and thrived, many rural Georgia counties have struggled, prompting residents to migrate to urban centers like Atlanta and Savannah. Small towns, especially in South Georgia, sometimes have more shuttered storefronts than open businesses. In recent months, lawmakers have even considered offering tax incentives to people willing to move to rural areas.

White Georgians in places like those are among the main targets for the hard-edge ads with anti-illegal immigration themes ads that were prominent in the Republican primary for governor. One candidate rode around in a “deportation bus,” promising to fill it with “murderers, rapists, kidnappers, child molesters and other criminals.” Another, Secretary of State Brian Kemp, bragged that he had a pickup truck big enough to “round up criminal illegals and take them home myself.”

Mr. Kemp’s homespun, hard-right ads (in another, he is seen brandishing a shotgun at a young man who wants to date his daughter) probably helped lift him into a runoff with Lt. Gov. Casey Cagle, the Republican front-runner.

Concern about illegal immigration runs high in the state, especially among conservatives. A poll of likely primary voters, commissioned in April by The Atlanta Journal-Constitution and WSB-TV, found that nearly 82 percent of Republican respondents rated the issue as “very important” or “important” to determining how they would vote, as did about 60 percent of Democrats.

Liberals often see such concern over illegal immigration as a malign response to the rapid changes in the state. Census figures show that the white share of the population is down to 61 percent, from 72 percent in 1980, while the number of Hispanic residents has surged. Statewide, more than 1 million people now speak a language other than English at home.