The abrasives you will need to use are either boron carbide (quite expensive) or diamond (even more expensive). Boron carbide is very messy to use, but should do the job. Follow the basic instructions for vibratory tumbling outlined in How to Polish Stones. You should not need much in the way of cushioning agents, as corundum is approximately as tough as diamond. If your stones are already somewhat smoothed, as rubies or sapphires recovered from a streambed might be, you can start with Step Two. Otherwise, start with coarse grit and go through all the usual tumbling steps, allowing lots of time for each one. To be sure you are ready for a finer level of grit, you can use a 10x loupe to make sure the scratches from the last step have been scoured away.
To polish your corundum, you are best off using diamond powder, although it is quite expensive. A very fine grade of boron carbide might also work, though it would be slower and would require some experimentation.
Tumble-polishing corundum is more of an art than a science, so if your first results are not quite what you hoped for, back up to an earlier step and try again!
Information from How to Tumble Polish Rocks into Gems by Edward E. Smith and various other sources.
Tumbling corundum?ruby and sapphire?can be done, but requires special abrasives (not to mention perseverance). The first thing to keep in mind is to not do this to gem quality material! The good stones would be much better served by being cut, as tumbling removes a lot of material in a rather uncontrolled fashion. A second point is that, while you can tumble-polish corundum in a rotary tumbler, this will take a very long time. We recommend using a vibratory tumbler for this reason.