I’ve never heard/read anywhere that Scots “lived in Ireland before coming to Scotland”; the history simply isn’t that old (Ireland is one part of Europe with the most recent human inhabitation). The Celts were steadily pushed across Europe from east to west, and the island of Britain was settled by Celts before Ireland.
Yeah, but different Celts from the Gaelic speakers. They spoke Brittonic languages, as opposed to Goidelic like Gaelic. They were Picts, British, Welsh. As for Scots living in Ireland before coming to Scotland, look at Dal Riada.