Does each new lord/lady of a vassal family have to swear fealty to their overlord, or is it explicit continuation from their predecessors? Does each new overlord or monarch summon their vassals to swear fealty to them upon ascension?

Good question!

The short answer is that it depends on the particular feudal culture and the particular period, because it matters whether the fiefdom was seen as the personal gift of the monarch or the rightful inheritance of the son, and so on. 

In most feudal systems I’m aware of, however, there was a necessity to have the new vassal make homage to their new overlord. It’s an opportunity to redo the ceremony – which is handy for the sovereign because it means that they get to do the ceremonies regularly and display their power. It’s also an opportunity for revenue-generation – a lot of feudal systems assessed a one-time tax when a fiefdom was inherited. And the same goes for a new overlord – you want to make sure that all the vassals know who’s in charge now, and to firmly establish their direct loyalty to the new man.

Moreover, if you think of it in symbolic/ideological terms, it makes sense that you would want to renew whenever there’s a transfer of power between generations. Remember, medieval politics is all about the body – the king is physically annointed, they give status and title by touching you or giving you a ring, and political influence comes from how close you are physically to the king. 

So if you have a new body or bodies standing in as vassal or overlord, it’s a new physical relationship and you need to enact the ritual again.