Days before the session, the commission’s de facto leader, Kris Kobach, the secretary of state of Kansas, had written a column for a far-right website noting that thousands of New Hampshire’s registered voters had out-of-state driver’s licenses. From there it was a short leap to the column’s startling conclusion: The narrow Democratic victory in the state’s 2016 race for the United States Senate appeared to have been “stolen through voter fraud.”

Matthew Dunlap, the Democratic secretary of state of Maine and a member of the commission, had been the most vocal skeptic on the panel. At the hearing, at St. Anselm College in New Hampshire, he called Mr. Kobach’s statement reckless. “That would be almost as absurd as saying if you have cash in your pocket that you robbed a bank,” he said.

Weeks later, Mr. Dunlap would sue his own commission, contending he had been illegally shut out of its deliberations by its Republican leaders. Shortly before Christmas, a federal judge agreed, and ordered the commission to turn over to him a sheaf of internal documents. And on Wednesday, barely a week later, the White House issued a brusque two-sentence statement disbanding the commission altogether, and parceling out its duties to the Department of Homeland Security.

The commission’s many critics applauded Mr. Trump’s decision. But his decision to hand responsibility for the investigation to a cabinet agency seems likely to spawn an entirely new set of battles.

To those who saw the inquiry as a partisan attempt to suppress Democratic votes in the guise of increasing election security, the decision shifted the search for evidence of voter fraud from a public commission to a federal law enforcement agency with less transparency than any blue-ribbon commission.

Mr. Kobach, who plans to remain an adviser to the inquiry, said he wanted Homeland Security to match its databases on legal and illegal immigrants with state voter registration lists to weed out the noncitizen voters that he and Mr. Trump claim are well hidden and ubiquitous on voter rolls.

But the department must take specific legal steps to obtain the state data, and any effort to simply hand over the commission’s voter data would be met with a lawsuit, said Jon Greenbaum, the chief counsel at the Lawyers’ Committee for Civil Rights Under Law. The civil rights group has so far prevailed in a federal suit that forced the voter fraud commission to turn over evidence indicating that some commission members were excluded from internal deliberations.