The controversy over kneeling in protest of racial injustice moved beyond the world of professional sports this week, when a number of schools told students they were expected to stand during the national anthem.

On Long Island, the Diocese of Rockville Centre, which runs a private Catholic school system, said students at its three high schools could face “serious disciplinary action” if they knelt during the anthem before sporting events.

Sean P. Dolan, a spokesman for the diocese, said on Friday that the letter, which was sent to principals, was intended to restate policy that the diocese already had in place.

But he added in an emailed statement: “Although the Diocese does not agree that demonstrations are appropriate in its schools during the playing of the National Anthem — which recognizes the tremendous sacrifices of Americans of all races, ethnicities and religions — it notes that students who seek to challenge racism and racial discrimination are firmly in accord with Catholic teaching.”

In northwest Louisiana, Scott Smith, the superintendent of schools in Bossier Parish, said student athletes were expected to stand for the anthem. “It is a choice for students to participate in extracurricular activities, not a right, and we at Bossier Schools feel strongly that our teams and organizations should stand in unity to honor our nation’s military and veterans,” he said in a letter obtained by The New York Times.

Waylon Bates, the principal of Parkway High School in Bossier City, La., a municipality of more than 60,000 people near Shreveport, outlined the punishment students would face at his school. He sent a letter on Thursday to athletes and parents saying athletes were required to stand “in a respectful manner” during the anthem.

An image of the letter was posted online by Shaun King, a journalist at The Intercept.

“Failure to comply will result in loss of playing time and/or participation as directed by the head coach and principal,” the letter said. “Continued failure to comply will result in removal from the team.”

A call to Parkway High School on Friday was referred to the school board, which did not comment on Mr. Bates’s letter.

In the past week, National Football League players have demonstrated during the national anthem in a show of solidarity against racial injustice and President Trump, who scolded the league and its players for protesting. Colin Kaepernick, the former quarterback for the San Francisco 49ers, spent much of last season either sitting or kneeling during the anthem, inspiring student athletes to recreate the gesture last year. But Mr. Trump’s involvement has given the debate new momentum and helped push it beyond the sports world. This week, military veterans, actors and the singer Stevie Wonder took a knee.

When the issue is swept up into the public school system, as is happening in Louisiana, it runs up against students’ First Amendment rights and a Supreme Court ruling in 1943, which said public school students could not be forced to salute the American flag or recite the Pledge of Allegiance if it conflicted with their religious beliefs. That ruling involved a case of Jehovah Witnesses who were expelled from school for not reciting the pledge.

“The law does not permit schools to forbid students from expressing their views, and all schools should be on notice that these policies are in fact unconstitutional,” Marjorie Esman, the executive director of the American Civil Liberties Union of Louisiana, said in an interview on Friday.

The Supreme Court also touched on students’ right to peaceful protest during public school hours in 1969, when it ruled in favor of students who wanted to wear black armbands to protest the Vietnam War.

The 1969 ruling essentially said that students do not lose their constitutional rights to free speech at the schoolhouse gate, said Francisco Negrón Jr., the chief legal officer of the National School Boards Association. If a protest is not disruptive, public schools have to allow it.

“That is the challenge for school districts in this scenario,” he said in an interview. “They have got to balance that with the rights of students to protest in a way that is not disruptive.” He said some school districts are “opting to use this as a teachable moment.”

Private schools that do not rely on government funding have more flexibility in setting their own rules for student behavior. The Diocese of Camden in New Jersey said last year that any student who failed to stand for the anthem at a sporting event would be suspended for two games. Repeated offenses could get students dismissed from the team.

“The best approach is helping our young people understand that blood was sacrificed so that we all can enjoy the gifts of our faith and our country,” the diocese said in a letter to its schools, NJ.com reported this week. “However, let me be clear. We are not public institutions and free speech in all of its demonstrations, including protests, is not a guaranteed right.”

Ms. Esman said it was “troubling” that Louisiana school administrators “seem to not understand what they are supposed to be teaching their students — the right to protest peacefully.”

The protests are “a statement about racial justice in this country,” she added. “The fact that there are so many people who are publicly saying they are concerned about this means it is a problem that needs to be addressed.”