DAYTON, Ohio — Perhaps nothing better encapsulated the tricky calculus of running for president in the midst of a natural disaster than the mixed messages flashing on a giant screen here Tuesday at an event for Mitt Romney.

At one moment, the screen featured a public service announcement instructing crowd members to text a $10 donation to the American Red Cross on their cellphones. The next, it displayed a glossy campaign video describing Mr. Romney as “charismatic” and “authentic.”

In a corner of the high school gymnasium where the event was held, people lined up to deposit boxes of bottled water and bags of brown rice for victims of the storm. In another, a woman proudly held up a T-shirt that read “Obama, you’re fired.”

The storm that ravaged much of the East Coast has pushed the presidential campaign into a delicate and ambiguous phase. Suddenly, it was a sideshow, with a caveat: Americans were still watching and assessing an improvised leadership test for both candidates as they navigated the politics of a deadly storm.