You could definitely hand down some of your magic items, and you can also inherit the older generation's contacts and allies. If the two groups overlap time wise, higher level characters can pass down gear they have outgrown. You might also consider some variation on the Weapons of Legacy. Even if it was just a Longsword +2 when it was introduced, taking up your father's sword/staff/mask (or having him give it to you after you prove that you are ready for it) can be a pretty powerful narrative moment.

One thing that could be fun is to run at least a few sessions alternating between the old crew and the new. If we are talking literal parents and children, the parents are probably still late 30s/early 40s when their kids are ready to start adventuring. Maybe set up a few situations where two things need to be dealt with simultaneously, so that the Old Guard handles the higher CR mission while the Youngbloods simultaneously handle the lower CR (but still important) mission at the next session. When you are ready to take the training wheels off, maybe engineer a Glorious Last Stand TPK for the Old Guard.

If the Old Guard just retires, what sort of benefit they provide would depend on what they are doing in retirement. The fact that they've decided to stop crawling down holes looking for treasure doesn't mean they are sitting in their rocking chair. If they are any sort of lord, some of their authority would be implied for their progeny within their own domain. If a character invests his money to start a merchant house, he will have easy access to transport and contacts all over the place (and if there is political intrigue, being employed by a merchant house gives you a good cover story for poking your nose into all sorts of places "as potential investments"). Someone who secures a court wizard gig would have access to high society, as well as access to the Royal Libraries. Any of these characters could provide adventure hooks.

Basically, just think of all the advantages privileged people have IRL, and translate it into game terms.

If there is a clear 1:1 connection between. Old and New, I might set a rule that nobody can create a character that is their own offspring/trainee. Let them RP being each other's mentors. That allows them to interact with their mentors without without having to play both sides of the conversation or having the DM take over a PC.