There is no comparable population movement in European history. The Muslim Iberian (and Sicily/Southern Italy) states were principally case of elite replacement not mass migration like we are seeing now, mostly it was in cities that new population arrived and they left a relatively limited genetic legacy in only a few areas. The Steppe invaders likewise were mostly just a ruling caste who basically disappeared into the much larger genetic milieu that looked largely the same before and after their period of dominance.

Modern mass European migrations are true population changes comparable only to something like the colonization of the Americas, except occurring at a much faster rate (it still took hundreds of years for Mexico and South America to become "mixed"). Nor has there really been anything comparable to the level of integration these new populations quickly achieve - the religious cultural and linguistic barriers are much less severe than most historical examples.

In continental history the closest event is probably the movement of Anatolian/Near Eastern farmers mixing with Europeans as semi modern farming spread. Again that's something that took place over probably hundreds of years though and those groups were much more closely related (especially at that point) to Europeans.