The genetic point is interesting. Modern historical/archaelogical work seems to be tending more and more towards the "cultural diffusion" model, as opposed to old"big population movement/invasion", until post-Roman empire, say. This changes post 300AD, so it is currently supposed that the subsequent changes did indeed involve substantial population movement(Anglo-Saxons into England, Irish Gaels into Scotland, and probably Britons into Brittany, to quote the relevant ones here). The implication is that Ireland and Britain had a fairly stable genetic pool up to around 300AD, going back many thousand years.
   Now, it is becoming established that are genetic differences observable between Wales, ireland and England, and between the West of England and the east. The interesting corollary to this is that as the AnglSaxons were coming from the Celtic heartland of Europe (genetically speaking) into an area with a substantially unchanged "old population" which had acquired Celtic cuture and language by a diffusion process. not by substantial invasions of people.
    So the latest position, which seems quite well documented and becoming more so, is that genetically speaking the English are the ones with the "Celtic" genes, and the Irish and Welsh aren't. Funny, eh?