This came up with my latin professor and he said that various frescos in Pompeii show some similarities, but he, not being an historian, wasn't qualified to answer. So I thought I'd ask here. I looked through the Popular Questions but couldn't find a satisfactory answer. Thank you in advance for the reply.
This is a hard question to answer because the root of the answer is pretty contested by historians. I suppose the best way to answer it would be to compare remains and look for notable differences. I am unaware of any attempts to do that, so I'll tackle it more on the level of population movement.
Ancient Rome was perhaps the most diverse city of its time. Juvenal (satire III) likes to poke fun at the minorities in the city. It was full of Greeks, Jews, Gauls, Iberians and just about any other group from the Mediterranean not to mention the other ethnicities in Italy itself (Etruscans, Oscans, etc.). So I guess I can answer your question by calling into doubt that there ever was such a thing as an "average citizen of Rome."
But I won't. The city was pretty badly afflicted during the age of barbarian migrations. It was sacked three times in the fifth century, then besieged twice in short order when Justinian tried to reconquer it. By the time of Justinian's war (535-554) the city of Rome was already pretty depopulated (see Procopius, Gothic War 2). This means that later migrations, especially the Lombards at the very end of the Gothic War, could have an outsized effect on the population.
This raises a controversy about how many barbarians actually invaded the Roman empire. The problem is that we have only a few numbers and they may be less than reliable (Procipius claims that 80,000 Vandals invaded Africa. Were they all ethnically Vandal? Did that included the women and children? Is he exaggerating for effect?). Therefore historians are divided. Some believe the migrations were pretty small and probably had little effect on the local population. Others argue that they overran the native population and began a huge change.
If you begin to take into account all of the population changes in Italy (invasions from Normans, France and the Holy Roman Empire, migrations throughout the peninsula, the re-population of Rome), then it seems like there is a lot of room for sharing genes in Rome, but it is hard to know whether that actually happened.
I'm in Naples just now and there are people on the streets who look just like the marble statues in the archaeological museum.