Obviously "being in circulation" is different from merely surviving. I have a sizeable collection which is not at all synchronic. So let's start from there, let's assume swords were being passed down for lengthy periods of time. Would said swordsman want to use an old sword?

There's the problem of value. If you've kept a sword long enough to be noticeably "out of date" there is probably some sentimental value attached to it. If you have the training which makes using a sword feasible then clearly you've a) had access to training gear and b) are able to get hold of a newer sword.

Then there's the problem of efficiency. A migration era sword could survive the introduction of the longsword. But do you really want to go up against something like a longsword with a single edged blade from a German forest? Doable, sure, but why risk it?

Obviously these are extreme cases. The truth of the matter is periodization is a shaky business, the lines become more blurred the closer each period is to another. Lines aren't hard and fast and certain weapons become out of date more quickly than others. Also it's never as if each culture has a single contemporary style but a myriad of variations subject to yet further change.

Maybe pick up something like Oakeshott's "Record of the Medieval Sword" to have a look at some changes over time.

It's not uncommon to see swords being used for several centuries in the early middle ages (esp. between the 5th-7th centuries).

There are several ways to see this in the archaeological material. Artistic styles change over time, and many swords have been found in graves which have hilt fittings that belong to one century, and scabbard fittings which are of a style not seen until 100 years later (suggesting that the scabbard was replaced or exchanged on a much older sword - or at least, a sword with an older hilt). Weapons of an older style are also sometimes found in graves with other objects (like coins) from centuries later, meaning the sword was likely an heirloom when it was buried. And some weapons show a lot of wear.

Early medieval swords had a lot of separate parts (hilt fittings, etc), and it's not uncommon to see these pieces removed from the swords (like in the Staffordshire Hoard). Sometimes, older pieces would end up on newer weapons, or old weapons would have some of the older pieces removed and replaced. This is probably evidence of the making over of an old (and possibly famous) weapon to be re-gifted to a new owner. This can often be spotted on the archaeological finds.

But it gets harder to track after the 7th century, because (in most cases, except in Scandinavia) people stop being buried with weapons, and the archaeologically recovered weapons tend to be found in rivers. Without a context, it's much more difficult to date these weapons precisely, and it's a lot harder to tell how old they were when they were disposed of.

Also, you start to see (in England, at least) laws requiring the family of dead men to pay a 'heriot' or death tax of weapons to their lord. Presumably, they're supposed to give the lord back the weapons he gave the dead family member, so he can give the weapons to the new heir when he swears his oath of fealty. There's not enough evidence to say what the effects of this would have been though - does this mean that heirloom swords were surrendered to the lord, or did families have their own weapons that didn't get returned? Would the lord give a son his father's sword, after it was returned to him? There are similar practices in France developing around the same time.

This answer's pretty France and England heavy, and really only deals with the early middle ages - I'm not sure, for example, how the Scandinavian evidence compares, or what changes in the late middle ages. I know late medieval Scandinavian literary sources talk about swords that are passed down through many generations (there's one great saga where a daughter robs her father's tomb to get his cursed magic sword back so she can keep fighting a feud).

The best article if you want to know more about swords in the early middle ages is a chapter by F. Theuws and M. Alkemade in Rituals of Power (Brill: 2000).