While agreeing with Ross’ general argument in this post, I feel compelled to say something about this:
That being said, culture certainly can be passed on through conversion – otherwise Christianity wouldn’t have survived Goths and Franks and Vandals…
A better example might be the conversion of Slavic peoples to Christianity, from both Catholic and Orthodox sources, since the Goths and Vandals were already Christians (albeit Arians) when they settled in Italy, Spain and North Africa, and in any case the numbers of all these groups were nowhere near the scale of replacing or outnumbering the local Gallo-Roman or Roman population. Actually, the Goths and Vandals provide support to Ross’ larger claim that the best means of reproducing culture is through inheritance received by one’s children, since as distinctive groups the Goths and Vandals were obliterated by the Byzantines and their cultures and languages had not been adopted by their former subjects. The examples of the Goths and Vandals also serve as a useful reminder that the religious culture of the majority will not necessarily follow that of the ruling elite, and that religious culture is more enduring than changes in regime.
Then, of course, the vast majority of the Christian world in the east never had to contend with any of these peoples anyway, so Christianity was going to survive there even if the various Germanic peoples had been far more numerous and had remained pagan. At the time of the barbarian migrations, the overwhelming majority of Christians was in the east, and the east remained the center of the Christian world for several centuries after this.