I'll be the first one to sound the 'correlation is not causation' alarm over and over. After all, maybe the social issue is getting enough people to protest because society is already on the verge of change, rather than society changing because of the protests. Protesters may be getting the causation backwards. However, there's some evidence that protests are helpful in causing change.

My favorite study to establish a directional causal link looks at policies enacted after protests, accounting for the effect weather has on protests. The weather is presumably random enough to not be related to where society is, and it affects protest turnout, which lets you connect protest turnout to policy change causally. It's not definitive, but yeah, there's decent evidence.

I disagree with the other poster who is suggesting that marching is a necessary step for change, but it does seem to be a useful instrument for it.