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.”

However, she 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.

Judge McCafferty was appointed by former President Barack Obama. The Trump administration is expected to appeal her ruling.