It sounds like you are talking about conversions during the post-Roman period, so I'm going to stick to that.

During the early history of Islam, there isn't much of a history of effectively converting pagan religions. The areas invaded by the Rashidun and Ummayad Caliphates were pretty much all populated by Christians and Zoroastrians. People generally converted to Islam to escape the differential legal and social treatment of non-Muslims. The speed of this really depended on the state of the existing religions. Christianity in Egypt was very strong, and the Christian community remained the majority for centuries, and still persists today. In contrast, Christianity in North Western Africa was a complete disaster at the time of the conquest and was effectively gone after a few generations.

Conversions to Christianity in Europe happened for a variety of reasons across time and space.

There were many nations that converted to Christianity effectively as a top down political decision, examples of this are Bulgaria, Hungary, and the Scandinavian countries. They wanted to integrate themselves into the European state system at the time. This was relatively "easy" in the sense that it usually happened over a couple of generations, but there was organized resistance from pagan nobles in some places, like Bulgaria and Norway

Other parts of Europe were converted through violence. Charlemagne subjugated and Christianized the Saxons in a series of brutal wars. The Baltics were converted in a series of crusades mostly in the 12th and 13th centuries.

I think the structure of the faiths at the time demanded expansion, unlike some other faiths, but the conversions happened for a variety of reasons including 1) High political reasons 2) Force 3) Individual belief 4) Individual self-interest.