“He got into a Twitter war with the president of Mexico,” she added, alluding to Enrique Peña Nieto. “He is temperamentally unfit and totally unqualified to be president of the United States.”

Mr. Trump, in turn, addressed Mrs. Clinton’s own qualifications at an event in Virginia Beach. Asked if questions about Mrs. Clinton’s use of a private email server as secretary of state made her unfit to serve as commander in chief, Mr. Trump replied with a quip.

“It’s such a long answer to that question,” he said to laughter. “Could go on for days.”

Mr. Trump will also use a speech on Wednesday in Philadelphia to call for the end of the military sequester as a way to increase defense spending, according to a senior adviser who spoke anonymously in order to discuss campaign strategy. The speech is also intended to contrast Mrs. Clinton’s foreign policy vision — which Mr. Trump described on Tuesday night as “military adventurism” — with what he says is a more cautious approach.

On Wednesday night, Mrs. Clinton and Mr. Trump will answer questions in a televised forum on national security hosted by the Iraq and Afghanistan Veterans of America.