Beshear responded that he supports his children and told Bevin that he would "never attack yours" in one of several testy exchanges between the rivals who are locked in a tight gubernatorial race. It reflected their animosity toward each other, built over their terms when Beshear filed several lawsuits challenging some of Bevin's executive actions or legislation he backed.
The high-stakes debate came three weeks before their Nov. 5 election showdown.
The candidates wrangled over health care, pensions and support for coal miners, but the hourlong debate reached into Beshear's family life during a question about education.
Beshear touted his proposed $2,000 across-the-board pay raise for public school teachers and vowed to submit "an education-first budget" to lawmakers.
Beshear has made education a centerpiece of his campaign, pointing to Bevin's feud with teachers who opposed his efforts to revamp underfunded public pension systems.
In response, Bevin fired back at Beshear: "You love public education so much that your kids go to private schools, is that right?" Beshear responded: "You're attacking my kids now?"
Bevin said public education needs to be funded but ridiculed his opponent's teacher pay raise proposal, calling it "Andy Candy." During the campaign, Bevin has accused his opponent of failing to lay out how he'd pay for a teacher pay raise and other proposals.
Afterward, Beshear told reporters that Bevin's performance was "unhinged and erratic" and "showed that he is not fit to be the governor."
"I'm not surprised that this governor attacked my kids tonight because he's been attacking all of the commonwealth's children through his policies these last four years," Beshear said.
Bevin said his comments were not meant as an attack on his opponent's children.
"I did not bring up his children, I brought up a decision he and his wife made on behalf of their children," the governor told reporters. "The children did not make the decision to go to private school."
Bevin took aim at Beshear's family during another exchange over a question about support for higher education. Beshear pointed out that his father, Steve Beshear, was a "poor preacher's kid" who was able to attend the University of Kentucky and went on to become a two-term Kentucky governor.
"If you have to brag about how poor your father and grandfather were, that's a bit of a stretch," Bevin replied.
On the health care front, the candidates sparred over Bevin's efforts to require some "able-bodied" Medicaid recipients to get a job, go to school or volunteer to keep their benefits.
The governor defended his effort, saying his objective is to ensure that the medically frail and disabled "do not lose out" on coverage. A federal judge blocked the work requirements and Bevin's administration is appealing. Medicaid is a joint federal-state health care program for the poor and disabled.
Beshear responded that Bevin's proposal would result in at least 95,000 people losing coverage. The Democratic challenger said that if elected, he'll rescind Bevin's Medicaid proposal.
Beshear's father used an executive order to expand Medicaid coverage while he was governor. His order increased Medicaid rolls by more than 400,000 people, many getting coverage for the first time. The federal Affordable Care Act gave states the option of expanding Medicaid coverage.
Bevin has said the Medicaid expansion was too expensive.
