What about using a piece of a sponge? You can cut it to size so it doesn't take up a lot of space and then you can place the “stone” there to dry after you use it.
That is a great suggestion on how to dry it out in between uses. However, my biggest concern is how to take the irregular pieces and wet them, apply them and not drop them any further. Some way to enclose them and still have it porous enough that the substance can be applied. Something similar to a panty hose substance but much smaller. Sorry I didn't make myself clearer the first time.