You mean western parts?

As to your question, it's impossible to say. Certainly the landholding class became enmeshed with the older senatorial class that dominated landholding in the north, but we have no basis for guessing the extent of this process. As for the rest of the population, there probably a little bit, but again, it's so difficult to guess.

Edit: Jeez, how am I being downvoted. I know this sub bans speculation but the original question is unanswerable. This is a conclusion that I have surmised after a good amount of research about the early medieval Italy, especially the Franks and Lombards, and the answer to this question will always be that it is unanswerable.

Wickham takes stabs at what society looked like in, *Early Medieval Italy: Power and Society, 400-800, but he can't really say anything about the extent that local Italian society genetically absorbed the Lombards et. al.

Walter Pohl hardly addresses the topic in the only relevant chapter in, The Oxford Short History of Italy.

Even Neil Christie's, The Lombards can't get at any sort of answer. He talks about trade. Both Christie and Wickham talk about how the rulers who stepped in during the period were predominantly based in urban centers but he neither really talk about how they shaped the genetic makeup of the region.

Perhaps Wickham addresses the question in an offhand way:

The mixture of peoples is the best context in which to deal with two problems about continuity in the history of Italy. The first problem is that of the break in the sixth century: of how far Lombard occupation destroyed the social structure of Italy [not how much of a genetic footprint they left behind], pressing the bulk of the Romans down into the lowest ranks of the peasantry, and forming a tabula rasa on which history, as it were, began afresh. The second question, inextricably related to the first, is that of the continuity of the patterns and personnel of the aristocracy throughout all the political changes in Italy. Both these problems have often been posed as being momentous shifts in Italian society. These do not seem to me to have occurred [emphasis mine]. All through the early medieval period, the basic social structures of Italy and the patterns of its aristocracy were so firmly related to the economy that intrusions of new peoples did not make much difference to them.

Wickham guesses, but he's talking about something more concrete, something that archaeology can begin to tell us, what social structures looked like, not the extent to which the Lombards, et. al. changed the makeup of Italian population.

You might look at the types of collections in the Museum of the Early Middle Ages in the outskirts of Rome, which includes a wide variety of artifacts from Lombard Italy, but how much these artifacts say about population changes is difficult to say.

Well, the fall of the Roman Empire is quite an complicated issue; it was a gradual process that happened over centuries, and some institutions continued going strong for hundreds of years after, such as the economic infrastructure, where others, such as the army, deteriorated. That Germanic tribes caused the end of the Roman Empire (I'm not sure if OP is referring to the Western or Eastern Empire, but I'm assuming Western) is a bit of a myth and only a small part of a much bigger picture. I don't usually recommend Wikipedia articles, but the one on the Fall of the Western Empire is actually reasonably good at least in the sense that it gives a good idea of the timescale and complexity of the process, and introduces some classics interpretations behind the reasons for the Fall. Mary Beard in her new book SPQR for example says that such a thing as the Fall of the Roman Empire didn't really happen at all, seeing that the Roman civilization continued for centuries after the fall of the Western Empire in the Byzantine, and that lot of the Roman institutions lingered on under the Byzantine Emperors.

During the gradual decline of the Western Empire, there were a great number of different armed conflicts with the Germanic tribes, so I'm not sure which specific invasion or tribe OP is referring to. The city of Rome, for example, was sacked 3 times in Late Antiquity: 410 by Visigoths, 455 by Vandals, and 546 by Ostrogoths. Obviously, much more melees were going on in the northern part of Italy. I'm not sure how big a portion of the Germanic invaders actually settled in Italy; someone with better knowledge of the era may chip in. But if we were to study the genetic lines of modern Italians, I don't really know how we could differentiate the genes of these late antiquity Germanic invaders and between those Germans who migrated to Italy during the long centuries when Germanic provinces were part of the Roman Empire.