Christianity, on the other hand, was obviously spreading mostly with active conversions - at first, in early Roman times, through teachings of the Apostles and their continuators, then in late Roman times in a state-sponsored way (it became the state religion of the Empire), and later during Medieval times also in a state-sponsored way (monarchs were Christianizing their subjects) or in an otherwise imposed way (like for example conquest by crusaders).
I think that during the Roman Empire much of the spread of Christianity within the empire was actually down to population movement rather than conversion. There isn't much evidence for substantial conversion during this period, but attacks against Christians were sometimes motivated in part by the fact that they were foreigners. E.g. the famous persecutions in Lyons in 177 were directed against Christians who had migrated there from Asia Minor, one of the major Christian population centres. Remember that at this stage Christianity was a primarily Greek-speaking religion even in the western empire, because most western Christians were actually easterners.
Also, don't forget that later, in the fourth and fifth centuries, Christianity spread throughout the Sassanids' empire and on into central Asia primarily through forced population movement. The Sassanids captured huge numbers of Christians from the Roman Empire and resettled them in empty territories of their own.