> Sometimes peacefully, like Anglo-Saxons pushed out local Britons in England

The Battle of Chester has entered the chat.

No one ever "peacefully" pushes anyone else out of their homes.

Naah, I'm pretty sure history can find an example of pretty much anything. Here's another example (just less researched) -- Slavic dispersion in Europe. And I'm 100% sure there a lot of other examples of peaceful expansion or assimilation, because for the most history land was abundant, but shortage of hands to tend to it. There was just no reason to fight (and record it).

It doesn't mean some kings declared that land their own (some declared everything), but they couldn't enforce it. So usually it boiled down to main argument whether something is "yours" -- collecting taxes (aka tribute). As long as someone's can enforce collecting tribute, then they deserve to have the title of "owning" it.

Btw, One of the ways historians determine whether migration was mostly peaceful is by looking at archeological gender structure -- if there's a lot of female immigrants (e.g. Slavic expansion), then they are more likely to be moving by whole families. But if there are few females -- more likely it was invasion (e.g. Huns). Not absolute signal of course, as nothing is in history.