Not so sure that’s the case in Eastern Europe. The actions and words of the Visegrad Group suggest otherwise.
An excellent point I'd like to expand on if I might. What appears to be the case is that the Czechs don't want their country to be overrun, nor the Slovaks, Poles, Hungarians, or lately the Austrians and the Greeks. But nobody much is objecting to Europe as a whole being overrun, which argues both a failure of a collective European identity (that was one of the major aspirations of the EU) and a certain stubborn ideological faith either that Europe will be bettered by a forced demographic change or deserves it for past sins.
Certain Swedes, for example, have very publicly proclaimed that there is no Swedish culture. The level of denial necessary even to make such a statement reflects the degree to which zealots are willing to reject their own culture which they don't understand in favor of a foreign culture that is yet to be formed, a new culture which very obviously does not contain certain fond assumptions of which equal rights for women is merely the most blatant example. A new culture under a new demography is hardly likely to sustain old human rights assumptions, and if they won't be defended they won't exist.