I would be quite interested in who these "many" Frankish and Gothic nobles were. Of course, in the year 500, for all intents and purposes the Roman Empire still stood, so you might have a slightly different situation. However, by 550 and especially by 600 this would have been quite rare. It's cerainly commented upon, such as in the lives of Holy Fathers of Merida, where the arrival of Greek merchants warrants a mention in the text c. 550 CE. The people who did travel were usually ecclesiastics (I'm thinking here of figures like John of Biclar, or the future Pope Gregory I). Most people were resolutely local and would have stayed so. If you read, e.g., Gregory of Tours, there is very little sense of movement even to other places in the West, with the exception of the occasional campaign against a neighbor. Perhaps someone would go on a pilgrimage to Rome. Perhaps someone might even make it to Constantinople. One guy--Willibald--definitely made it to the Holy Land in the eighth century, but that's pretty rare.

The best book for this is Michael McCormick's Origins of the European Economy 300-900, where he tracks every single journey made during the period 700-900. He gets to about 700 people if memory serves, plus some additional pieces of evidence for movement like coins and relics. On the one hand, it's a lot. On the other...compared with the Roman world, or the high medieval world, it's almost nothing. And those are all primarily nobles (ambassadors), churchmen, and slaves.

Educated is another question entirely. Again, probably limited for the most part to ecclesiastics, and the old senatorial families (by 600 in Gaul, basically the same thing). If you read German then check out Stroheker, Karl Friedrich. Der senatorische Adel im spätantiken Gallien. Tübingen: Alma Mater Verlag, 1948 and Heinzelmann, Martin. Bischofsherrschaft in Gallien: Zur Kontinuität römischer Führungsschichten vom 4. bis zum 7. Jahrhundert. Soziale, prosopographische und bildungsgeschichtliche Aspekte. Munich: Artemis Verlag, 1976.

Both make the point that the only place where the old Roman traditions were continuing was the old senatorial elite, which had become the bishops of Gaul by this time