Why do you believe that there was a sizeable migration? Where from and where to? Have archaeologists found changes in housing structures and burial customs that signal a change in population?
Edward Lluyd also spelled his name as Lhwyd. He was a Welsh scholar and patriot and became the Keeper of the Ashmolean Museum in Oxford. He studied the ancient languages of Ireland, Scotland, Wales, Cornwall and Brittany. He recognised that these languages had similarities and decided that that meant they were survivors of the ancient language of Gaul. Without being explicit, he announced that this group of languages was "Celtic" and published his views in 1707. No one know why Lluyd chose the name "Celtic".