Mark Dubowitz, the chief executive of the foundation and a self-described political independent, acknowledged to me that “after the election, some F.D.D. people wanted to join the ‘resistance.’ ” But that’s not what his organization does, he said. “Our mission has been to provide research and policy options to three administrations to defend American interests and global leadership.”

Access is one reason. Money is another.

Among the biggest donors to the Foundation for Defense of Democracies is Bernie Marcus, a co-founder of Home Depot and a major Trump supporter. In June 2016, Mr. Marcus wrote an article endorsing Mr. Trump and urging Republicans to get in line: “I have a message for the #NeverTrump crowd: Enough already. Donald Trump is our presumptive nominee and it is time to get over wishing it were not so. If you don’t, change your social media hashtag to #HillaryGOP.”

But it’s very difficult to draw a direct line between pro-Trump donors and the operations of a particular organization. In part that’s because all of the boards of these think tanks are mixed bags when it comes to Mr. Trump.

Take the American Enterprise Institute. Its large board includes Seth Klarman, a well-known anti-Trump billionaire, and Dick DeVos, the billionaire husband of Mr. Trump’s secretary of education, Betsy DeVos. A number of A.E.I. scholars have joined the Trump administration: Scott Gottlieb is now the commissioner of the F.D.A., and Kevin Hassett is the chairman of the White House Council of Economic Advisers. And yet the think tank is also home to Norman Ornstein, a liberal who just co-wrote a book about the danger of the Trump presidency. Though he says he is further to the left than almost all of his colleagues, he said that “I don’t get any sense at all that what they’ve been doing or what they’ve been saying has been shaped in any way by the donors to the institution.”

“I have little doubt that some of our donors rail against me,” he added. “But I’m not told about it. And if anyone in a position of influence here leaned on me politically I wouldn’t be here. I’d leave.”

From the outside, the board of the Manhattan Institute seems similarly diverse: The pro-Trump intellectual Roger Kimball sits alongside William Kristol, a leading Never Trumper. But Rebekah Mercer is also on the board, and several sources described her as “very active.” In his resignation letter, Mr. Stern singled out Ms. Mercer and her billionaire father, calling her “an accomplice in one of the most malignant political movements in the country.”

The Mercers are the key backers of Breitbart News: In 2012 they donated $10 million to the site, which Mr. Bannon has boasted is “the platform of the alt-right.” After Andrew Breitbart’s death later that year, Mr. Bannon took over the site, and under his watch it has become an open sewer of conspiracy theories and barely veiled white nationalism, anti-Semitism and racism. The Mercers also serve as patrons of the former Breitbart editor Milo Yiannopoulos, a man whose views have earned him a proud Nazi salute from Richard Spencer. (Just today, however, Mr. Mercer announced that he was “severing all ties” with Mr. Yiannopoulous, saying he was “mistaken” to have supported him. He also said he was selling his stake in Breitbart to his daughters.)