There is no doubt that the demography is changing in Europe.
In so far as Africa and Asia is concerned, there is no doubt that there is a concerted effort by both the Christian and the Muslims clergy to increase their band of followers.
To this end the last Pope's statement of harvesting of souls and also the spread of Saudi Wahabism is of note.
On the issue of culture and tradition, one wonders if globalisation is the real catalyst that is nibbling away at the 'old culture' (whatever that means).
I don't think the Europeans expect that immigrants will shed their culture as soon as they touch the European shores. But maybe they feel uncomfortable when some section of the immigrants want their culture to supersede the local culture. It is the same angry feeling that many exhibit here about evangelism being thrust in India or the attempts by Wahabis to convert or the massive influx of illegal Bangladeshis into India and changing the demographic pattern in the border areas of Assam and Bengal.
Imperialists of yore could impose their culture on the colonies because they were the rulers. However, the immigrants are at a disadvantage since they are not rulers, most of them being economic refugees and at the mercy of the country they have emigrated to.
It is like forcing Indian religious rites and law in Muslim countries where the Indians go for work. They live their lives there and yet they cannot become citizens of the country. And they are subject to Islamic Laws like the Sharia.
I think it is a very complex issue that has pro and cons on both sides.