Like other constitutional rights, the ones guaranteed by the First Amendment are not absolute. They may conflict with other rights, such as the right to a fair trial, or be in conflict with powers given the state and federal governments, such as the police power to preserve the peace or the power to effectively make war.

Marches, picketing, rallies, and even pamphleting have historically been subject to time-manner-place restrictions. Government may not restrict speech or assembly based on what is being said, but they can restrict when and where it occurs, to protect the public. In the mid-century South, allowing a civil rights march almost inevitably meant there would be a counterprotest by whites who preferred the status quo and often became violent in making their arguments. So even leaving aside the prejudices of the government officials who had to grant parade permits, it was indisputable that allowing a civil rights march would make for a rough day in town and long shifts for the police.

The usual three-step process was this: deny the civil rights march a parade permit. When the march occurs anyway, invoke the "riot act" by announcing that the assembly is unlawful and must disperse. The third step was where things got ugly in the 1960s in the South, as white police and bystanders either didn't allow the marchers any time to disperse, attacked them as they did, or arrested those who for tactical reasons chose not to comply with the order.

You may be interested in reading the Supreme Court case Cox vs. Louisiana, 379 U.S. 536 (1966) both for the facts of how that Baton Rouge march played out and the legal analysis.