WASHINGTON — With their nomination contests finally locked up, Hillary Clinton and Donald J. Trump sought to rev up their core supporters on Friday during dueling speeches in which each cast the other as unfit for the presidency.

Mixing policy arguments with personal attacks, the presumptive nominees clashed from across the capital as they previewed the arguments they will take to the country over the next five months.

For Mr. Trump, who suffered perhaps the most damaging week of his yearlong campaign, a forum for evangelical Christians proved to be an unlikely haven, while Mrs. Clinton was greeted with cheers and pink “Hillary” signs as she vowed to protect women’s rights from Mr. Trump and the Republicans.

“Anyone who wants to defund Planned Parenthood and wipe out safe, legal abortions has no idea what’s best for women,” Mrs. Clinton said of Mr. Trump at a Planned Parenthood Action Fund event here. “After all this is a man who has called women ‘pigs,’ ‘dogs’ and ‘disgusting animals.’”