What probably happened was this: for some unknown reason, the proto indo-europeans in russia (or turkey, depending on which origin theory you like better) made the switch from a matriarchal to patriarchal society and gradually became more aggressive and warlike. over thousands of years, small groups of indo-european warriors (as in dozens or hundreds) left their tribes to conquer their peaceful neighbors, and set themselves up as a warrior aristocracy. Because of their superior technology (horses and chariots) and the indigenous europeans’ peaceful cultures, they were probably able to do this relatively easily (not unlike the Conquistadors in early modern times) They didn’t leave much trace in terms of DNA because there were so few of them.

Once they had subjugated an indigenous tribe, they intermarried with them and over time, the whole tribe adopted their masters’ language and customs. Over thousands of years, this phenomenon snowballed, though it probably wasn’t the main vehicle for indo-european expansion. The indo-europeans likely had a higher material standard of life than the natives, thanks to their propensity for raiding and for enslaving or patronizing skilled artisans. after centuries of skirmishing and trade, some of the leaders of the indigenous tribes probably thought the indo-europeans had the right idea, and began adopting their language and culture voluntarily – eventually, this trickled down to the commoners as well. this process kept spreading westwards until nearly all of europe considered itself indo-european; of course over time, these assimilated tribes gradually formed subgroups that took on their own identities (Celtic, Italic, Germanic, Slavic, etc…)

The irish seem to be one of these indigenous groups that at some point adopted Celtic culture, and may very well have originated in Spain. There is an Irish legend concerning Míl Espáine (king Milesius), who spied the coast of ireland from his watchtower in Galicia and led the first Gaels to settle there. This legend may be a Celticized version of an earlier indigenous one describing how the aboriginal irish settled the island – if that’s the case, it might explain the genetic similarities between the irish and atlantic iberians.

One more interesting note – celtic languages are divided into two types: p-celtic and q-celtic. P-celtic languages include Gaulish and the Britonnic dialects, while q-celtic languages include Goidelic (Gaelic) and Celtiberian – the language of the spanish celts. Again, more evidence for a common ancestry.

Also, one element that were are forgetting is the extent to which the culture and traditions of pre-Celtic Ireland blended with the ways of the incoming Celts. The Celtic identity adopted by the Irish probably absorbed a good amount of the autochthonous culture, which would explain in part the nuances that differentiate the “Gaelic” or “Goidelic” Celtic culture and language of Ireland from the separate Celtic cultures and languages of ancient Britain and mainland Europe. Sure, some of the difference can be explained by drift and isolation over time, but there are probably many subtle elements, undecipherable to us today, that remain in Irish culture from the pre-Celtic customs, whatever they looked like.

Similarly, we should remember that the “Celtic” peoples of continental Europe, too, at some point adopted Celtic languages in place of older tongues. Ethnic Celts didn’t replace the previous inhabitants of Gaul anymore than ethnic Romans and Franks replaced the Gauls themselves… Just as the Gauls started calling themselves French and speaking the Vulgar Latin dialects that evolved in Romance languages, their ancestors probably picked up Gaulish from their neighbors at some point.