PHILADELPHIA — Donald Trump’s effort to throw Hillary Clinton off balance by saying Russia might be in possession of thousands of her deleted emails had an unexpected effect this week, after Clinton’s team used it as an opening to lob their own national security arguments against Trump.

It was a bold move for Clinton, as the scandal surrounding her use of a personal email system when she led the State Department has caused her several headaches, and has led to polls showing voters’ trust in her has eroded. Still, her team gambled that they could use Trump’s argument to try turning the tables on the issue.