In the case of Italy, the first barbarian ruler is Odoacer. We don't know much about him (he was probably a memebr of the Scirii, some say he was a Hun). In 476 he deposed Romolus Augustus and proclaimed himself "King" or 'Dux" of the germanic peoples in Italy. He already was the commander of the German troops stationed in Italy (primarily Herulians), so what he did was essentially a coup. At this point the civilian population of Italy was still made up by Romans.

Then Odoacer was replaced by Theodoric, king of the ostrogoths. Theodoric was sent to recover Italy by the Byzantine emperor Zeno, but in practice he ruled the territory as its King. He brought with him 20,000 warriors and perhaps 80,000 civilians. Keep in mind that the Roman population of Italy consisted at its peak in the 1st-2nd century of 10 million people (some say 6 millions, others say 14 millions) with the city of Rome having one million (again, estimates are complicated). Around 500 AD, population was MUCH much lower but still there several millions. So what you had was effectively a minority of Goths ruling a majority of Romans. The even smaller number of germans part of the army of Odoacer, after being defeated, possibly joined Theodoric.

Then you have the Greco-Gothic war from 535 to 553, where the Byzantines manage to reconquer the Italian peninsula. Despite the fact that they were Eastern Romans coming to recover the West, from the Italian point of view this was THE real barbarian invasion of Italy. An army of Greeks/Syrians/etc fighting against the Gothic army essentially caused a significant drop in population, bringing destruction and epidemics etc etc.

Then in 568, the Longobards descend in Italy. And they capture several areas of it from Byzantium (most focus on Lombardia Major, consisting primarily of Northern Italy). How many are they? Again, probably something like 100-150k like the Goths. Except now the Roman population of Italy in late 500s is way smaller than the early 500s. The Lombards however are still a minority of rulers among the Romans. A more sizeable one though, which was also joined by the previous Goths.

The Lombards rule varying areas of Italy until the Franks of Charlemagne conquer them in 774. From 568 to 774 a lot happened. The Lombards created their own set laws, that gradually replaced in use that of the Romans. They picked Latin as their language and abandonded Aryianism in favor of Catholicism. And the Romans, and here is the answer to your question, abandonded their name. If you went to a village of northern Italy and asked "who are you" in 700 AD the paesant would have told you "me? I'm a Christian" and then he would have probably said "a Lombard" (because he most likely followed Lombard law).

So, in short, by 800 the Romans in northern Italy had mixed with the Lombards and other germanic peoples. The people in Rome, which usually resisted the conquest from all these peoples, continued to call themselves Romans. The people around Ravenna too (the region of Romagna), again because it was not conquered by the Lombards but primarily because it was the heart of Eastern Roman administration in Italy. In southern Italy you have certain areas of Romans mixed with Lombards like in the north. And also the areas under Byzantine rule, where old Latin romans are mixed with the Greeks who came from the East.