There is typically a top-down political component to cultural shifts like you're asking about where governments chose the religion for the people. Additionally, times of political chaos make people more willing to accept new belief systems. The many civil wars of Rome facilitated the adoption of Christianity before it became the state religion. Similarly the political instability at the collapse of the Han Dynasty helped Buddhism gain acceptance. A happy afterlife is especially inviting in such tough times. Otherwise, when empires or kingdoms actively patronized a different faith than they had before, they did so for political reasons. In ancient Egypt, Akhenaten promoted a small time cult over the traditional religious figures to weaken their priesthoods. Even the Protestant Reformation's success was due to the patronization of local leaders in the Holy Roman Empire. Henry VIII of England couldn't get the pope's approval for an annulment so he made a new church. I'm sure there are counter-examples of bottom to top religious changes, but prior to religious toleration, the church and the state were typically hand in hand.