Properly updated clients will contact the checkpoint server and request the current correct status of the network.  The client will compare the answer it receives with the information it currently sees, and correct itself if necessary.  A client may be on the wrong fork for a block or two in such a scenario.

Presumably the majority of people will update their clients properly.  All of the "lynch pin" clients will undoubtedly do so (exchanges, pools, etc.).  The rest of us will follow suit because we are keen to have DRK which are actually usable.  

Since there is only one checkpoint server orchestrating the whole show, everyone on the correct client gets the same information and we can all carry on business as usual.

Those who didn't update their clients and end up on the wrong fork will (I assume) simply continue mining worthless DRK until they pull their heads out, get it together and update.  As long as they don't make up more than 50% of the network, they're irrelevant.