I do this and have done it since my earliest playing/DMing days. If my player characters or my player's characters lived long enough often they got married and/or had children. Possessions, heirlooms, birthrights, legacies, and inheritances went to their offspring, brothers, or sometimes to their friends, fellow party members, or more distant relatives.
Lands, estates, livestock, valuables, etc. were handled in the same way. Written wills were rarely used because usually everyone knew who the inheritor would be. Blood relatives or someone the dying party member had already publicly announced as his heir.
A couple of things to keep in mind about this from my point of view. As DM how the players handle these issues is their affair, as has already been mentioned. Unless the law gets involved and I have to arbitrate or play judge. Magical items and devices in my setting are rare, so no new character is likely to inherit a balance tipping cache of magical goods. They are simply too rare and powerful. Each magical item in my setting is unique (there are no generic +1 longswords) and so how each magical device or item functions also tends to be unique. Meaning that just because it is inherited doesn't mean the new player knows how to use it, that it can be used or activated in the same way as the former owner used it, or that the device will even have the same powers or capabilities it previously displayed. Each item acts and reacts to each owner uniquely and individually.
This makes for great new adventures as new characters try to understand what exactly it is they have inherited and how it works. New PCs have also inherited famous items only to discover that the magical items are apparently no longer magical or have transformed themselves into something else entirely than was previously expected or suspected. And some new PCs are able to awaken wholly new capabilities from a magical item than was previously the case. And finally some characters have inherited family heirlooms and such thinking they were not magical (for previously they showed no magical dweomer of any kind) only to discover that for the new owner the item is apparently magical.
And of course claims can always be contested, both legally and covertly. Ambushes occur over valuable items, keeps are assaulted, estates attacked and burned, caravans raided. Just because you inherit something doesn't mean it happens securely. And this also makes for good adventures.
Then finally, some legacies are passed down but not given directly. The opportunity exists to recover a legacy but not the guarantee. The inheritor has to discover the location of a hidden treasure hoard, or must complete a quest to recover his inheritance. Worthiness to inherit must sometimes be proven. And of course there are also secret caches and secret inheritances that are not immediately obvious. Again a good genesis for an adventure, or a Quest.
I've often used heirlooms, legacies, birthrights, blessings, and inheritances to good effect in-game. Sometimes it even leads to absolutely great adventures.
P.S: I also forgot to mention, you can associate inheritances, legacies, heirlooms, birthrights, and items to be inherited with riddles, puzzles, and prophecies, as easily as with Quests. And those makes great adventures too.
