Modern scholarship recognises that Q-Celtic languages were not brought to Scotland from Ireland, rather, Q-Celtic languages were once spoken across the entire British Isles and while the successive phases of the adoption of P-Celtic languages never reached Ireland, they did reach Scotland but didn't entirely engulf it. The Pictish language most likely incorporated Q-Celtic, P-Celtic and pre-Celtic elements. The P-Celtic influence in Scotland didn't extend further than the Highland Line, north and west of which a language we would more accurately call Gaelic than anything else was spoken long before Fergus Mor mac Erc led the Scots to Dalriada at the end of the 4th century CE.
I wouldn't entirely agree with this...Some modern scholars may hold this view, but equally there are those that hold the view that distinguishing P and Q Celtic languages at such an early stage is incorrect, because the differences of the Ps and Qs developed and became pronounced at a relatively late date in the evolution of Celtic languages.
Your assertion about P Celtic influence not reaching beyond the Highland Line doesn't hold true if you accept the evidence of 'pit' placenames for one...and assuming any value can be found in agreeing that Pictish is indeed a P Celtic language like the rest of Britain was until the Goidelic language started gaining influence, as is commonly held. I'd really appreciate any references you might have on this to make sure my information is up to date

The Picts were not "shoved" by the Scots anywhere - the kings of Dalriada in fact occupied a subordinate role to the Pictish kings more often than not, and indeed Dalriada was completely overun by the Picts by the middle of the 8th century CE during the reign of the Pictish king Angus mac Fergus (the likely origin of the carved wild boar to be found at Dunadd, the citadel of the Scots of Dalriada - the boar was a prominent Pictish emblem, which indeed was the "Scottish" royal standard until the adoption of the Lion Rampant by William I "the Lion" in the 12th century CE).
The idea of any sort of invasion by the Dalriadans in wholesale terms is lacking in evidence, really. In fact, archaeologically the evidence suggests that cultural influence went from Scotland to Ireland in terms of the types of buildings we find in both areas...The issue of Pictish/Dalriadan politics is hazy at best, as indeed is the reason for the presence of the 'Pictish' boar at Dunadd. Ewan Campbell focuses on this in his book on Dunadd, along with an interesting article about the ogham script found there, which has been partially translated.