Even in the States... when I was traveling cross-country in the mid-1960s every area had its own regional flavor. The way a burger was made, the dialects, the stores, what people wore, etc. Made a similar trip in the mid-2000s and so much had changed. The interstates, for one, zipped us by the smaller towns (whereas the old highway system used to take us through them). The same chain restaurants were everywhere, the kids all looked the same, the clothes were the same, a lot of the really thick accents were not really there (except in older people). The music was the same. The TV in the motels was the same and there was a clone mall in every larger town. We saw new buildings going up that had the same architecture as SoCalif strip malls and apartments. It was, actually, kind of sad. (The move from a local culture to a national one. Although it is nice that some of each culture is now in the mainstream.)
But it was/is inevitable. It will be more inevitable as populations increase, the media saturates and permeates, and people move around more. And they will move. Europeans rode amazingly tiny boats to North America. In the Dust Bowl years of the 1930s in the USA people in the mid-west moved West (and were unwelcome) and rural types moved to the cities. Desperate people are now ~walking~ to the USA from South and Central America. As a rule, people go where they think they have a chance of a better life, rather than stay where the land (or weather/economy/politics/crime/neighbors/institutions) will try to kill them or keep them at barely subsistence levels.
Nobody can stop that kind of migration, unless they can ameliorate the cause(s). And I don't see that happening - there is, apparently, no will to do so. I can't see how it won't be just as in the States - people will come to Europe as long as the Middle-East is in turmoil and people are starving in Africa. The EU is going to have to sort out how to handle it. And when the EU hits capacity, it'll spill over to non-EU. Like a somewhat perverse version of Sim City. One can erect fences, but eventually the weight of bodies against the chainlink will break them down and it'll be a flood. Better to deal with it incrementially, I think.