I've read that well up until the time of Nero, Italians constituted the backbone of the legions but from there on out, provincials instead constitute the majority of the legion's man power. My question is, are many of these provincials the sons and grandsons of Italian legionaries? I've read that most provincials fell into this category, being that their mothers were often non-Italian - is that accurate?
I have seen you (or somebody?) post this question a few times now without answer, so I will write and say that I don't think there is a satisfactory answer. It might be possible to arrive at this answer (probably through inscriptional evidence), but I don't know how to find that data, if it has been gathered. I will talk a bit about the basics, though.
First, probably the best resource for this is Mann, Legionary Recruitment and Veteran Recruitment Under the Principate, 1983. I don't have access to that at the moment, however.
First, what you read about the decline of "Italians" in the legions after the 1st CE is probably correct. I think it is Mann who estimates that only 1 in 5 legionnaires were recruited within the geographical bounds of Italy by 100 CE. After Augustus, it was more attractive for the Italian potentials to join up with the Urban cohort, or the vigiles, or the Praetorians. Better pay, close to home, etc.
Second: the "Italians' in the provinces. In 30 BCE Octavian disbanded and settled the dozens of legions under arms that had accumulated in the last two decades of civil war. He founded nearly 30 veteran colonies inside Italy, and as many or more veteran colonies in the provinces, especially in Spain, Gaul, and North Africa. These three in particular would become population and economic powerhouses in the Imperial period. In Spain, Augusta Emerita (mod. Merida) became a very successful metropolitan center. In North Africa, the dusty ruins of Carthage were reinvigorated, and that city would become one of the four biggest in the whole Empire by the 2nd century; in Gaul, places like Vienne, Nimes, and Arles were urbanizing rapidly. There are dozens more I'm not mentioning, as well. Urbanization of the Mediterranean absolutely exploded in the 1st c. CE.
The veteran colonial settlements either seeded or augmented these places, but the veterans themselves were not the only population. The local populations were also participants and must have made up the bulk of the demographic. After a century and half a dozen generations, would a young man of Augusta Emerita be able to trace his great great great grandfather's identity as an old veteran settler? I'm not sure, and I'm not sure if he would even be interested in asserting that identity. I'm fairly sure we never see claims like that on funerary epitaphs (though I'd be happy to be wrong about that!).
As far as socio-economics are concerned, it would follow that at least some of those civil war veterans went to their colonies in the far-flung provinces and became important members of their communities. Probably not enough of them, however, to have a significant impact. Demographics of Imperial Roman provincial cities is definitely not my area, though; perhaps a core element of those original colonists persisted even into the 2nd c. CE? And if so (but I doubt it), would those groups be more prone to sign up for the legions? Lots of questions there.