The Koch political network remains generally opposed to many Democratic policies and does not want to see leaders like Nancy Pelosi, the House minority leader, return to power if the Democrats triumph in the midterms. But those who know Mr. Koch’s thinking said that his criticism of Mr. Trump reflects the vast political and personal gulf between the two men. They also echo the widely held beliefs of many conservatives who worry that Mr. Trump is inflicting long-term damage on their cause, and on the country.

Mr. Koch’s unease is a reflection of the wider discomfort and disorientation inside the Republican Party since Mr. Trump stormed the presidential primaries in 2016 and knocked out every candidate the Koch network had supported.

Mr. Trump all but ran against elements of the Koch agenda in 2016, promising support for entitlements, infrastructure and the military. In some ways he has governed like a conventional pro-Koch Republican, especially with some of his cabinet picks like Scott Pruitt, his former chief at the Environmental Protection Agency. But his language on trade and immigration ran counter to the Kochs. And like some other right-leaning organizations and intellectuals, Mr. Koch and his organizations have struggled to find their place in a political party they barely recognize and whose political fortune is now wedded, for better or worse, to Mr. Trump’s.

Despite the hundreds of millions of dollars that the Koch network has spent on politics in recent years, Mr. Koch has always been unnerved at seeing portrayals of himself as a Republican kingmaker, people close to him have said. And over the last few years the Koch brothers, two of the world’s wealthiest men, have tried to cultivate a worldly, civic-oriented image to counter the Democratic Party’s attacks on them as self-interested corporatist puppet masters. Koch Industries, the global energy conglomerate, also began a national marketing campaign to try to soften its image.