While some boards have grown very large, in part to increase the donor base, most are governed by just a handful of people at the top. On rare occasions, a trustee comes to define an institution, as Mr. Lauder has the Whitney.

Over four decades, Mr. Lauder, who is the elder son of the cosmetics impresario Estée Lauder and went on to run her company, has been the museum’s biggest donor, contributing large sums of money as well as major works of art. Although he has a deep affection for the Whitney’s former home, designed by Marcel Breuer, on Madison Avenue and initially opposed the idea of a move downtown, Mr. Lauder not only came around but donated $131 million to the new building in the Meatpacking District, which was ultimately named after him.

Even as chairman emeritus and well into his 80s, Mr. Lauder has continued to wield remarkable power at the Whitney, if lately on a less day-to-day basis. Mr. Griffin, when asked to explain his reason for deciding to stay on the Whitney board after resigning, provided a statement that made it sound as if Mr. Lauder was still the man in charge, and made no mention of the museum’s current leaders. “I have great admiration for Leonard Lauder’s leadership in building the Whitney into one of the greatest institutions of American art in the world,” the statement said. “I will continue to support Leonard’s vision for the museum, and am proud to serve with him on its board.”

From board member to broad target

In 1996, Mr. Kanders, a former money manager who had ended up in the eyeglass and lens business, bought part of a company called American Body Armor. It was his first stop on a journey that would remake him into a provider of law enforcement equipment and military supplies. Today his company, the Safariland Group, makes a wide variety of products including bulletproof vests, bomb-defusing robots and gun holsters.

But it was one product — tear gas — that labeled him Museum Enemy No. 1.

Late in 2018, at a time when President Trump was warning of an illegal immigration “crisis,” United States Customs and Border Protection agents used tear gas on asylum seekers at the Mexican border, firing into a crowd that included children. The website Hyperallergic reported that the canisters had been manufactured by Mr. Kanders’s company.

Days later, about 100 Whitney staff members, ranging from front desk personnel to senior curators, signed a letter. “We work to bring in artists who are immigrants and artists of color to the collection,” it said. “Upon learning of Kanders’s business dealings, many of us working on these initiatives feel uncomfortable in our positions.”