Religious beliefs can combine with social movements when those movements adhere to the religious beliefs but religious beliefs and traditions are not always synonymous with ethics or morality. The case can be made in any religion as none of them are fallible: Christian persecution of homosexuals, Muslim repression of women. When religious traditions refuse to change with changes in social evolution, those traditions can impede social transformation from religious and political standpoints. The continuing debate of the separation of church and state (in America) is evidence of this problem. Clearly, this separation is not absolute since many politicians loudly proclaim their religious beliefs and God is printed on American currency and even invoked in the Pledge of Allegiance. The separation is to prevent politicians from allowing their religious beliefs from interfering with legislation and policy, which is directly tied to social transformation.
I agree with what the first poster said about religious support during the Civil Rights Movement of the 1960s, but religion was also used as a means to pacify slaves in America with the lesson that while they may suffer in this life, a greater reward awaited them in the afterlife. So, in this case, religion was used first to impede social transformation and in the second to augment it.
If you use the term "religious tradition" to refer to the ethical beliefs of a religious group, then religious traditions can contribute to social transformations by pushing a society to conform more closely to the values that are implicit in the religion's ethics. This can be seen clearly in the role that religious groups have played in such things as the Civil Rights Movement in America.
The Civil Rights Movement used the language of Christian religion to achieve social transformation. It appealed to white Christians to live up to Christian ethics by treating black Americans as their sisters and brothers. By appealing to the values that were implicit in Christian teaching, the movement was able to achieve a major change in American society.
When a society's practices are (or potentially are) in conflict with the ethics of a major religious tradition in that society, the religious tradition can help to push the society towards acting in a way that is more in line with those ethics. In this way, a religious tradition can contribute to social transformation.