Granted that it wasn't a dio .... I re-did a figure base not too long ago and it really wasn't too difficult.
When you say things are "connected to the base" are you talking about just connected to the clay or does it go deeper than that? Here's two ways I'd go about it...
1) Gently pull up on the figures, etc until they dislodge. Depending on how strong they are you might just want to pull at the feet. If you take some papertowel and wrap it around the piece that you're going to pull on you can protect it. Plus, if you use something like needle-nose pliers, you can use them parallel to the base, using a wooden dowel or th base itself as a fulcrum. This will give you a lot more control.
2) If the above doesn't work, just try tearing up the groundwork from around the exsisting pieces. This will allow you to peck and chip away at the groundwork from around whatever you used to attach the figures, etc. Then either cut the attachments, or go back to "above".
HTHs.