Donald J. Trump easily dispatched his Republican rivals in the Michigan and Mississippi presidential primaries Tuesday and won the Hawaii caucuses, regaining momentum in the face of intensifying resistance to his campaign among party leaders.

Senator Bernie Sanders scored an upset win in the Michigan Democratic primary, threatening to prolong a Democratic campaign that Hillary Clinton appeared to have all but locked up last week.

Mrs. Clinton lost badly in Michigan among independents, showed continued weakness with working-class white Democrats, and was unable to count on as much of an advantage with black voters as she had in the South.

Addressing reporters in Miami while the votes in Michigan were still being counted, Mr. Sanders said that his powerful showing indicated that “the political revolution that we’re talking about is strong in every part of the country.”