Well for starters, a blockchain is simply a decentralised ledger and as such, data that's been put on a blockchain is expect to remain there in case of future reference so I'm pretty sure nothing would happen to the data. It would just remain untouched until someone comes to try and pull up some data from it.
I imagine there'll be some way of accessing the information and pulling up data from the blockchain, and depending on who's updating it, you may even find a final entry into that particular blockchain as "Deceased owner" or final details about the person that owned it.
Whatever the case, I believe that the blockchain and all its information about the person will remain in tact so that other people will be able to come and pull up information, if necessary, from there.
I hope this helps.
What happenes with current data? Lays around till its deleted by someone. Except with blockchain stuff, as long as the chain has active nodes, it'll stay there, forever.