Hillary Clinton invoked her roles as mother and grandmother on Saturday to deliver an impassioned rebuttal to Donald J. Trump’s contention that her push for stricter gun control would make families less safe, saying the presumptive Republican nominee would put more children “at risk of violence and bigotry.”

The day after Mr. Trump received the endorsement of the National Rifle Association, Mrs. Clinton assailed her probable general election rival as pandering to the group.

“I believe it’s the most powerful lobby in Washington,” Mrs. Clinton said of the N.R.A. at an event in Fort Lauderdale, Fla., to benefit the Trayvon Martin Foundation’s Circle of Mothers. “And we know some candidates will say or do anything to make them happy.”

Speaking in a ballroom full of mothers who had lost children to gun violence, Mrs. Clinton defended her position on gun control and her promise to overhaul the criminal justice system.