Every case is different. But if there is a link, it will usually be through a single individual, not some kind of mass migration of half a clan. Often that person will be an obscure younger son of a younger son. If you're lucky, you'll be able to pin him down. Mostly you won't.
The only place to look is at the bottleneck. You'd need to figure out when and where the Trizzinos first appeared in the south. This may not be where they were most numerous later.
Once the branches have separated, they're separate. Even in Italy, you don't need a family feud for people to lose touch with their 2nd cousins, long-distance or not. You wouldn't expect to find evidence at later periods.
Random connections may exist that don't go through the gateway bottleneck, but they don't tell you anything about anybody else, so they don't answer the question.
Bear in mind that the connection will be shared by a lot of people and you won't be the first person to look.
There's always a story, but that means nothing. If it turns out to be true, somebody's grandad got lucky - he didn't know it was true when he was making it up.