The Buffalo News editorial board called her “frankly, wacky.”

At an early voting site in Amherst, a Democratic-leaning suburb, a handful of voters said on Friday that they had voted for Mr. Poloncarz because of his management during difficult emergencies, including the pandemic and a blizzard that killed 31 people last year. The office of county executive is responsible for leveling taxes, maintaining infrastructure and providing other essential — and often apolitical — services to residents.

Garrett Hodgson, 48, a Democrat, said the migrant crisis was “not something that’s in Poloncarz’s control.” Sharon Carlo, 78, a former Republican who recently registered as a political independent, said the Republican ads focusing on the accusations against Mr. Poloncarz were “none of our business.”

In Suffolk County, an area more populous than 11 states, Democrats say they see signs that the Republican fever-pitch of 2021 and 2022 has cooled, particularly around crime. But strategists in both parties said Republicans were still well positioned to reclaim the county executive seat, putting them on the brink of a remarkable takeover of almost every major office on Long Island.

“Each cycle builds into the next,” said Jesse Garcia, the architect of Republicans’ multiyear strategy to hit Democrats, who control Albany and New York City, over the state’s bail law, rising prices and, increasingly, their handling of the migrant influx.

Edward P. Romaine, 76, the longtime Republican supervisor of Brookhaven, the county’s largest town, is running for county executive against Dave Calone, a Democrat and former prosecutor and businessman who has never held elected office.