A federal judge in New Hampshire limited on Thursday the Trump administration’s ability to withhold federal funds from public schools that have certain diversity and equity initiatives.

The judge, Landya B. McCafferty, said that the administration had not provided an adequately detailed definition of “diversity, equity and inclusion,” and that its policy threatened to restrict free speech in the classroom while overstepping the executive branch’s legal authority over local schools.

She also wrote that the loss of federal funding “would cripple the operations of many educational institutions.”

The decision followed a demand earlier this month by the Trump administration that all 50 state education agencies attest that their schools do not use D.E.I. practices that violate President Trump’s interpretation of civil rights law. Otherwise, they would risk losing billions in Title I money, which is targeted toward low-income students. About a dozen states, mostly Democratic leaning, refused to sign the document.

In issuing her decision, Judge McCafferty declined to issue a nationwide pause on the policy. Instead, she limited her ruling to schools that employ or contract with at least one member of the groups that brought the lawsuit: the National Education Association, the nation’s largest teachers’ union, and the Center for Black Educator Development, a nonprofit that seeks to recruit and train Black teachers.

The N.E.A. has about three million members, including some in states that bar teachers from collective bargaining. It is possible that most school districts in the nation would be affected by the ruling.