WASHINGTON — This summer, Attorney General Todd Blanche boasted about the drop in federal crime while stumping for the Republican candidate for New York governor at a rally headlined by President Donald Trump.

In Michigan, Sean Duffy, the transportation secretary, personally endorsed Mike Rogers, the Republican Senate candidate, arguing that Rogers would help advance his department’s work.

And in Alaska this month, Commerce Secretary Howard Lutnick stood with the president at a rally in which he noted the role of Sen. Dan Sullivan, a Republican, in securing a natural gas pipeline.

Such appearances have become commonplace as nearly all of Trump’s top deputies fanned out to promote Republican candidates facing difficult election odds.…