Minor population movements, sure, but there's no evidence of massive migrations from outside the Greek world or of population replacements. As for legend, there are many classical-era legends of migrations from one part of the Greek world to another; none about migrations from outside.

We don't know what people in the Mycenaean palace culture called themselves or their language, but the modern English name for their language is 'Greek'. There's no good reason to call their ethnicity anything other than 'Greek' as well. The fact that we use a separate term for the Mycenaean palace culture is just a product of periodisation, like 'Hellenistic', or 'Byzantine', or 'modern'. It isn't by way of indicating a change of language or ethnicity.

So 'evolve' wouldn't make much sense in this context.

Now, there were theories floating around in the 18th-20th centuries about migrations from outside the Greek world, but they're ethnic nationalist fantasies designed to cast the ancient Greeks as a Germanic-Nordic 'race' with blond hair and blue eyes, and modern Greeks as a separate 'race' (or even as a separate species). Here's a piece I wrote offsite earlier this year that goes into the history of the idea, and how it ties in with 19th-20th century fantasies about Atlantean/Hyperborean migrations.