Well the most common and effective way of destroying local cultures, is to force the people to speak the common language.
For example in France, the UK, and many other places, there used to be many local regions, with their own languages and strong local cultures and loyalties. The rulers wanted to kill the local cultures, so that the people would have no local identity. This stops disloyalty or independent thinking or independence movements.
They did this by forcing their subjects to speak a common language.
This policy was perfectly effective.
A province speaking its own language can easily maintain its own identity and push for independence. Without its own language this is more difficult, even if it keeps its own customs.
I think I've been too vague. So I can elaborate about these policies in the UK or France, if you like. For other territories (Spain, Italy, etc) I believe the same thing happened but I'm not the expert.