A coalition of 19 states sued the Trump administration on Friday over its threat to withhold federal funding from states and districts with certain diversity programs in their public schools.

The lawsuit was filed in federal court by the attorneys general in California, New York, Illinois, Minnesota and other Democratic-leaning states, who argue that the Trump administration’s demand is illegal.

The lawsuit centers on an April 3 memo the Trump administration sent to states, requiring them to certify that they do not use certain diversity, equity and inclusion programs that the administration has said are illegal.

States that did not certify risked losing federal funding for low-income students.

Rob Bonta, the California attorney general, said at a news conference on Friday that the Trump administration had distorted federal civil rights law to force states to abandon legal diversity programs.

“California hasn’t and won’t capitulate. Our sister states won’t capitulate,” Mr. Bonta said, adding that the Trump administration’s D.E.I. order was vague and impractical to enforce, and that D.E.I. programs are “entirely legal” under civil rights law.

The Trump administration did not immediately respond to a request for comment on Friday evening.

The administration has argued that certain diversity programs in schools violate federal civil rights law, which prohibits discrimination on the basis of race, color and national origin in programs that receive federal funding.

It has based its argument on the Supreme Court’s 2023 ruling ending the use of race in college admissions, arguing that the decision applies to the use of race in education more broadly.

The administration has not offered a specific list of D.E.I. initiatives it deems illegal. But it has suggested that efforts to provide targeted academic support or counseling to specific groups of students amount to illegal segregation. And it has argued that lessons on concepts such as white privilege or structural racism, which posits that racism is embedded in social institutions, are discriminatory.

The lawsuit came a day after the Trump administration was ordered to pause any enforcement of its April 3 memo, in separate federal lawsuits brought by teachers’ unions and the N.A.A.C.P., among others.

Mr. Bonta said that the lawsuit by the 19 states brought forward separate claims and represented the “strong and unique interest” of states to ensure that billions of federal dollars appropriated by Congress reach students.

“We have different claims that we think are very strong claims,” he said.

Loss of federal funding would be catastrophic for students, said Letitia James, the attorney general of New York, an adversary of President Trump who previously won a civil fraud case against him.

She noted that school districts in Buffalo and Rochester rely on federal funds for nearly 20 percent of their revenue and said she was suing to “uphold our nation’s civil rights laws and protect our schools and the students who rely on them.”

This article is part of our Museums special section about how artists and institutions are adapting to changing times.

Surrounded by a group of 10th graders, Alex Asal, a museum educator at the Science History Institute in Philadelphia, read aloud from three school lunch menus. She asked the students to raise their hands for which sounded best.

One menu had options such as pizza, Caribbean rice salad and fresh apples. Another had grilled cheese, tomato soup and green beans. The third featured creamed beef on toast and creamed salmon with a roll.

That menu — which did prompt a few raised hands — was from 1914, Asal revealed. A century ago, butter and cream were considered as vital as fruits and vegetables are today because the concern was less about what children ate than whether they ate enough at all.

The exhibition that had drawn students from the Octorara Area School District of Atglen, Pa., was “Lunchtime: The History of Science on the School Food Tray.” It examines how this cornerstone of childhood became deeply intertwined with American politics, culture and scientific progress.

From the earliest school food programs until now, “what’s been interesting for us about this topic is how discourses of nutrition and science have always been present,” said Jesse Smith, the museum’s director of curatorial affairs and digital content.

Smith didn’t anticipate just how timely the exhibition would be when it opened about a month before the 2024 U.S. presidential election. Robert F. Kennedy Jr., appointed secretary of health and human services by President Trump, promotes the removal of processed foods from school lunches. History shows that his isn’t the first attempt to change what people eat.

“Lunchtime” was developed from the Science History Institute’s collection of books and scientific instruments related to food science. Located just down the street from Philadelphia’s Independence Hall, where the Declaration of Independence and the Constitution were signed, the small museum and research library teaches the history of how science has shaped our everyday lives.

In 1946, President Harry Truman signed the National School Lunch Act authorizing the creation of the National School Lunch Program. According to the Food Research & Action Center, just over 28.1 million children participated in the school lunch program in the 2022-23 school year on an average day, with 19.7 million receiving a free or reduced-price lunch. In the 2023-24 school year, some 23.6 million students were enrolled in high-poverty districts that qualify for free lunch for all.

“It’s a service to students, and something we provide on a daily basis to help the students learn,” said Lisa Norton, executive director of the division of food services for the Philadelphia school district. “And we know that there are students that this is the only meal they are going to see.”

The exhibition opens with the 1800s, as industrialization brings people to cities, far from the source of their food. Producers would cut corners, mixing wood shavings with cinnamon and chalk into flour.

“Probably the most notorious example was the dairy industry, which routinely added formaldehyde to milk to keep it from spoiling,” Asal said.

And school medical inspections found that children were severely undernourished. Scurvy and rickets were widespread.

The Institute of Child Nutrition, at the University of Mississippi, maintains an archive of photographs, oral histories, books and manuscripts, and Jeffrey Boyce, the institute’s coordinator of archival services, provided several photographs for the exhibit. One shows a baby being fed cod liver oil, an old-fashioned remedy for vitamin A and D deficiency, in the age before vitamin-fortified cereal.

Philadelphia became one of the first cities to have a school lunch program and, over the next few decades, local programs spread across the country in a movement led largely by women. A federal response to school lunches would come from the National School Lunch Act.

“The National School Lunch Program is the longest running children’s health program in U.S. history, and it has an outsized impact on nutritional health,” said Andrew R. Ruis, author of the book “Eating to Learn, Learning to Eat: The Origins of School Lunch in the United States,” which Smith used as a resource for the exhibit. “Research in the ’20s and ’30s showed overwhelmingly that school lunch programs had a huge impact on student health, on educational attainment, on behavior and attitude.”

As farmers faced ruin in the wake of the Great Depression, the Department of Agriculture purchased surplus crops to distribute to U.S. schools and as foreign aid. This decades-old partnership made headlines in March when the U.S.D.A. announced plans to cut $1 billion in funding to schools and food banks.

School lunch programs have wide public support, but that has never stopped them from being a political football. In the 1960s, the civil rights movement drew attention to the fact that many poor children were still going hungry. The Black Panthers’ free breakfast program helped fill the gap and put pressure on politicians.

A table in the exhibition piled with Spam, TV dinners, bagged salad and Cheetos explained how military research into preservation created iconic American foods. These advancements, however, also helped put nutrition back under the microscope and led to the concern that young people were getting too much of the wrong kinds of foods.

The 1973 board game “Super Sandwich” tried to make nutrition fun, with players competing to collect foods that met recommended dietary allowances. Remember the controversy in the 1980s over whether ketchup qualified as a vegetable? It erupted in a larger battle over school lunch program cuts under the Reagan administration and further inflamed the national debate over school lunch quality.

The Healthy, Hunger-Free Kids Act of 2010, and the public health campaign for children by the first lady, Michelle Obama, resulted in more fruits and vegetables, more whole grains and less sodium and sugar on lunch trays. But balancing those regulations with what young people will eat is a challenge, said Elizabeth Keegan, the coordinator of dietetic services for the Philadelphia school district who advised on the exhibition. Especially when median lunch prices, according to the School Nutrition Association, hover around $3.

“We always say, for less than what you pay for a latte, schools have to serve a full meal,” said Diane Pratt-Heavner, the association’s director of media relations.

Following their tour, the Octorara students reflected on the tales of wood shavings in food. They debated the quality of their own school lunches and what they would prefer: more variety, more vegetarian and vegan options, less junk food.

“It made me feel like we should get better food,” said Malia Maxie, 16. “When she was talking about 1914, like how they got salmon — we don’t get that anymore.”

Those from generations raised on rectangular pizza may see it differently.

“From the days when I was in school, the meal program has totally transformed,” said Aleshia Hall-Campbell, executive director of the Institute of Child Nutrition. “You have some districts out here that are actually growing produce and incorporating it in the menus. You have edamame at salad bars. They are trying to recreate what kids are eating out in restaurants and fast-food places, incorporating it from a healthier level.”

Everyone has memories of school lunch. Boyce remembers “the best macaroni and cheese on the planet” and the names of the cafeteria ladies. Smith remembers the Salisbury steak and that distinct cafeteria smell. For Ruis, the best day of the year was when his Bay Area school had IT’S-IT, a local ice-cream sandwich with oatmeal cookies.

“So much has changed, standards have changed, and what is considered healthy has changed,” Keegan said. “But something that has never changed is that feeding kids a nutritious meal is important.”