As an Italian (and probably old enough) I can confirm, what made the "transition" to Italian was school but much more than that radio and television where an "official" italian is usually spoken.
In some regions (both in the north and in the south) many people are - still today - functionally bilingual, i.e. they still use largely dialect (and that is their "native" language) and switch automatically to Italian when dealing with (say) authority/government officers/foreigners or when they travel outside their region.
But I believe that at the moment the majority of people from these regions are in the "reverse" mode, i.e. they normally speak Italian but switch to the dialect in case of need or with older relatives/known friends.