I'm going to keep this short, because I can't type with one hand and it's past midnight.

In the case of the March on Washington, the roadways were stopped up as people entered the city, but not during the march. Due to the massive influx of people on buses, many roads had traffic jams. In one case, 450 (!) buses left from 143rd St Armory. The local government and police made sure that parking and other amenities were set up, as the Kennedy administration was in favor of the march and had instructed the government to fully cooperate.

On the day of the march, many roads were actually empty, just because people stayed home. Many government workers were given the day off, and tourists avoided the potential crowds and/or riots. In addition, certain important roads were closed around the area of the march, such as Arlington Memorial Bridge. Observers said that traffic control went off without a hitch.

Admittedly, this march was special due to the support of the government, but I suspect that similar scenarios played out in other marches. Generally, as the point was to be arrested the local governments knew protests were going yo happen and shut down the nearby streets. While I don't know how long they were unusable for, I would guess a relatively short amount of time.

Richard Weingroff is the DOT's unofficial historian, and wrote this https://www.fhwa.dot.gov/highwayhistory/road/road.pdf very interesting paper about the civil rights movement, which I referenced. See around page 157. I used Geoffrey Hodgson's book "America in Our Time" a little bit for the reaction of the Kennedy administration.