Smart locks like Slock.it are supposed to allow locks to be able to do things based on the blockchain state. Does that work with locks that don't have their own internet connection? If so, how does the lock make sure it has the correct version of the blockchain?
One question at a time:
Does that work with locks that don't have their own internet connection?
No, unfortunately blockchain consensus software always needs a working internet connection.
If so, how does the lock make sure it has the correct version of the blockchain?
Like any other node, the lock has to verify the blockchain state based on its underlying consensus protocol. Therefore it needs at least half a dozen connections to other ethereum nodes to make sure it does not get tricked by a malicious node. If all nodes agree the current state is the correct version, than there is provable consensus.
As I understand it, there are two pieces of the slock.it puzzle: the smart lock itself (many of which you can purchase today on Amazon or at your local electronics store) and the Ethereum Computer.
The Ethereum Computer is running a full node, which will stay up to date, therefore allowing any off-the-shelf smart device to (hypothetically) connect to it:
A tiny, preinstalled, preconfigured home server running both an optimized Ethereum node and exciting new decentralised applications, including Mist.
It seems like the Slock.it team is dedicated to making sure it is compatible with most of the standards being used by smart locks today. Furthermore, the aforementioned blog, mentions how "dumb" devices could work:
Any object that’s using ZigBee, Z-Wave, Bluetooth LE or Wi-Fi will likely be able to interact with the Ethereum Computer. When it comes to powering up a ‘dumb’ object, it will be a case of retrofitting it using smart plugs.