I think you are missing the point. The Iranian Farmers and the Caucasus Hunter Gatherers are not Iranians or Caucasians, they are ancient samples found from places called now the Caucasus and Iran and and may have given little ancestry to modern Caucasian groups or Iranian populations. Having affinities is not conclusive of actual ancestry. Also the farmers that went to Europe had livestock that did not originate in Anatolia but much further east. So how did they Anatolian farmers get their sheep and goats? By trading with other Neolithics, of course. And they mixed with each other, that is what humans usually do.
In my Southern European opinion, all these studies are biased by the geneticists' paradigms of what is indigenous, what is European, what is Middle Eastern, and Southern Europeans are always excluded from being the European crowd because they have a lower ancestry from the Eurasian Steppes, and greater Neolithic farmer ancestry. Look at David Reich's clines of European and Near Eastern Ancestry, most Europeans outside the Northern, Western and Central zones are excluded because of the so called extra Near Eastern ancestry. Anyway that is my two cents worth.
I agree with you except for the fact that Modern Iranians and Caucasians do indeed have a lot of Iran Neo ancestry, along with Anatolian farmer. The gene flow went both ways. The fact is that a wave of that ancestry spread into Anatolia and mixed with the Anatolian farmers living there. Perhaps through Anatolians it spread south into the Levant, eventually even reaching North Africa. It definitely spread into the Central and Western Mediterranean, again, as I've said over and over again, as an admixed group. There is absolutely no evidence from history of a movement to Italy or Greece, for that matter, from the Caucasus itself.
In terms of Italy in, say, the Bronze and Iron Age, I'm not sure if all of it came via the Greeks or if some of it came directly from Anatolia itself. (It certainly didn't go to Etruria, as the Etruscans clearly came from Central Europe, which 90% of the internet said was fantasy when I and a few others insisted on it here and on anthrogenica; one of the many things they and eurogenes got wrong about Southern Europe.) In isolated places like southern and southwestern Sardinia in the Iron Age, some of it may have come with Phoenicians. Perhaps a bit arrived in the same way in Northwestern Sicily. The rest of Sicily and the mainland are different.
The Moots paper, other than providing the ancient samples, tends to confuse rather than clarify the issue of what happened in Imperial Age. Once we get Southern Italian samples and Greek samples from the Iron Age, we'll know better how much Iran Neo/CHG, for example, arrived in Italy with the Greeks and then moved northward.
The problem with Moots is that it assumes every single burial sample is a long term resident of Rome, i.e. Roman or at least Italian. That's manifestly a simplification. Not every person who "looks" like a Levantine or even an Anatolian on a PCA would have become a long time settler whose progeny contributed to the local genomes. We can see that with some of the samples from the post Imperial period whom we've analyzed and who are manifestly northern European visitors to Rome. Had they done some isotopic testing we might have a clearer idea of who was "local" and who was not. Added to all this, in the period in question, some Romans still practiced cremation, so the sample is not representative.
Then, there's the question of the big demographic change even earlier than the end of the Empire. Rome was gradually abandoned as the seat of Empire. Everything shifted either to Constantinople or to Northern Italy. That's why the "tail to the east" ended. The traders left.
There is indeed also the period of the Germanic invasions. The problem with attributing much of the change to them, the popular back to the beginning scenario particularly in the north, is that every Germanic sample we've found is either I1 or R1b-U106. I can't believe that a paper purporting to deal with Italian genetics totally ignore yDna. The one thing it's really good for is tracking migrations. There's far too little of either, even in the Veneto, much less in Lazio, to account for a change from people with almost no steppe to people with 30% steppe. It doesn't matter how small the "native" population might have been; the "Germanic" ydna would have to be higher than it is. Not to mention that the Langobards numbered around 100,000 people even according to their own scribes, and the Goths were even smaller in number, mimicking what happened in Hungary.
I really hope the Reich Lab (and Razib Khan in his summary) doesn't make these kind of elementary errors.
Now, if someone shows the Germanics carried a lot of R1b U152 then that's a different story.
I'd also like to see samples from the Italian countryside and mountains from the Late Imperial and Post Imperial Era. When cities collapse, people from the periphery move down and repopulate them.
We need more data.