Songs can on occasion become important for a movement - "We Shall Overcome" is the classic case. It helped people in a very direct way when they were involved in direct confrontations with authioriy which needed confronting.

But most of the time songs that get saddled with the name "protest songs" aren't doing that at all. What they can achieve is get people to think for themselves and talk to each other, and that might lead them on to doing things together, and that's a movement. A quiet process often enough, and quiet songs can often do it.