As part of their longstanding crusade for lower taxes and smaller government, the Koch brothers in recent years opposed dozens of transit-related initiatives in cities and counties across the country, a review by The New York Times found. Campaigns coordinated and financed by Americans for Prosperity fought state legislation to fund transportation projects, mounted ad campaigns and public forums to defeat transit plans, and organized phone banks to convince citizens that public transit was a waste of taxpayers’ money.

By early 2017, Charles and David Koch, with a combined net worth of more than $100 billion, had become the leaders of a libertarian juggernaut loosely allied with the Republican Party, which, after eight years in the wings, again controlled the White House, both houses of Congress and many state legislatures.

Under the administration of Donald J. Trump, the Koch brothers’ prospects in Washington seemed improved, at least superficially. But beneath the surface lay substantive political and personal differences between the Kochs and Mr. Trump. While the Kochs did not endorse him, David Koch attended his election night victory party and later met with the president-elect at his Mar-a-Lago resort in Palm Beach. The Kochs contributed heavily to Vice President Mike Pence’s two campaigns for governor of Indiana, and counted a half-dozen close allies among the president’s cabinet choices and Republican advisers.

“The Kochs will be key figures in any discussion about what direction the party takes after 2016,” The Times reported in September that year, “and they are determined to steer it toward their free-market vision.”

That proved prophetic. As the 2018 congressional elections approached, the Kochs’ frustrations with Mr. Trump broke into an ugly and open exchange between Charles Koch and the president. Charles denounced Mr. Trump’s restrictive trade and immigration policies as divisive, and threatened to withhold the family’s support for Republican candidates who opposed the free-trade, government-shrinking policies at the heart of the Koch political philosophy. Mr. Trump struck back on Twitter, calling the Koch political apparatus “overrated” and “a total joke in real Republican circles.”

Politics and Profits

Critics accused the Kochs of buying influence and using their political machine to manipulate elections and government policies under a guise of patriotism and freedom. Those efforts, the critics said, cloaked an agenda to cut taxes and federal regulations governing business, the environment and other interests, primarily to benefit the Koch family and its enterprises.

Jane Mayer, the New Yorker writer and a critic of the Koch brothers, said in her book “Dark Money: The Hidden History of the Billionaires Behind the Rise of the Radical Right” (2016), that the libertarian policies they embraced benefited Koch chemical and fossil fuel businesses, which were among the nation’s worst polluters, and paid millions in fines and court judgments for hazardous-waste violations.