Between 1800 and 1860, religious people began to see social reform as inevitably related to Christianity. They began to see social movements as connected to the most vulnerable people, and thus as being a type of charity and "good action." This new way of thinking led many religous people to become involved in social and political movements. They were also motivated by the possibility of creating a new world order that was more closely aligned with Christian principles.
These groups were succesful to some extent. The abolitionist movement obtained an absolute victory after the Civil War. Movements that attempted to alleviate poverty or help groups such as orphans and widows also gained traction and support.
I think there are very few parallels with social movements today. While there are a lot of social movements nowadays, religious people are rarely a driving force in these. Moreover, when Christians become involved in a social movement, it is usually because it presents a direct question or threat to their personal beliefs.