Hillary Clinton and Donald J. Trump hopscotched from Pennsylvania to North Carolina to Michigan on Monday in the final, frenzied hours of the presidential campaign, offering clashing closing arguments as the sprawling map of the United States was reduced to a string of must-win states.

Accompanied by rock stars, ex-presidents, old friends and their grown children, the Democratic and Republican nominees pleaded with voters to end a traumatic campaign with an emphatic endorsement of their visions for the country.

In Philadelphia, Mrs. Clinton drew the biggest crowd of her 19-month campaign to the vast plaza in front of Independence Hall, where Bruce Springsteen, the balladeer of working-class America, rhapsodized about her values and the candidate portrayed herself as a protector of freedom and equality.

“Tomorrow we face the test of our time. What will we vote for — not just against?” Mrs. Clinton asked. “Every issue you care about is at stake.”