Harold, I think you have both items backwards. The Scots traveled to Ireland. Not the reverse. Why do you think The Romans built the Hadrian’s Wall? It was to keep the folks from Scotland from kicking their asses. There is a huge amount of history involved in this area. I suggest you read “Born Fighting” by Jim Webb, (Yeah, the (D) Senator) to get the results of his many years of research of this subject.
Maybe you’re right about the Celtic origins of the Picts. I did a quick Wikipedia search just now and couldn’t find any mention of some non-Indo European language. It might be a disproven old theory. I read a long time ago that the Romans described the Picts as being smaller and darker than other Celtic speakers and that they guessed that they had an Iberian origin.
As for the origins of the Scots, I refer you to SunkenCiv.