“You say something enough times, you make enough allegations, people start to believe it, and then they assume that there is rampant corruption,” Phil Runyon, the Peterborough town moderator, said, referring to voter confidence in elections at a Community Conversation event Thursday.

As distrust of elections roils national politics, Runyon and former Republican congressman Charlie Bass, now co-chair of the New Hampshire chapter of the Democracy Defense Project, delivered a blunt …