Mr. McCormick, Mr. Hovde and Mr. Sheehy will all face questions about their commitments to the states they seek to represent in the Senate. Mr. McCormick’s home in Connecticut was the main point of attack in 2022 when he lost the Republican primary to Mehmet Oz for a vacant Senate seat in Pennsylvania.

Mr. Hovde was raised in Wisconsin, attended the University of Wisconsin-Madison and counts Madison as his home. But his ties to California will be central to the Democratic case against him.

Mr. Sheehy appears to be a dream candidate for Montana, but in facing Mr. Tester, a flat-topped farmer from Big Sandy, Mont., his recent arrival in the state could prove to be an issue. He grew up in Shoreview, Minn., a suburb of Minneapolis, in a multimillion-dollar lake house, went to private preparatory school, and then to the Naval Academy in Annapolis, Md., before being discharged from the military with a Bronze Star and a Purple Heart. He moved to Bozeman, Mont., in 2014 and founded Bridger Aerospace and Ascent Vision Technologies, the latter of which he sold for $350 million in 2020.

Republicans involved in the general-election campaigns say they have plenty of issues to counter those charges with, at least to muddy the waters: a $1.3 million condominium in Washington, D.C., that Ms. Baldwin bought with her partner, Maria Brisbane, in 2021; the rising net worth of Mr. Tester; and family lobbying ties connected to Mr. Casey.

As for their standard-bearer, Mr. Trump, his scramble to come up with hundreds of millions of dollars in the coming days to meet the judgment against him for business fraud is raising questions not over how he made his money, but whether he can keep it.

His campaign, facing myriad financial pressures amid mounting legal bills stemming from the criminal cases against him, is scrambling to raise cash.

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