> I think the actual reason is something like a vast western empire promoting diversity as a good thing for your country.

The diversity train is a relatively new phenomenon.

EU expansion to the East had many factors. One of them being to have less barriers to get a cheap seasonal workforce pick vegetables in western EU countries [1].

Go back a few more decades: Germany in particular asked hundreds of thousands of people from Turkey to immigrate into Germany to help rebuild it after WW2. It's now the biggest foreign-origin population in Germany.

Not sure about France though.

[1]: https://www.nytimes.com/2017/12/11/business/europe-labor-rig...