Tumbling parts to achieve a deburred, deflashed or rounded edge is a common process. Both vibratory and barrel tumblers are used for this purpose. Choosing the proper media type is critical to achieving the desired finish.
Most applications require the use a preformed tumbling media. These media contain abrasives that are released into the tumbling process to remove the burr, etc. The (relatively) large mass of the media assists in this process by acting as a sanding block of sorts on the edge and surface of the part. After this abrasive process, parts can be tumbling in other media (including grits) to achieve a smoother or polished surface finish.
Tumbling parts simply with an abrasive grit such as Aluminum Oxide, Silicon Carbide or Walnut Shell does not typically achieve the deburring, deflashing or edge rounding required. While the grit may be very sharp, hard and abrasive, there is very little force acting against the edge of the part. The surface finish will probably be affected (roughening, smoothing or even polishing) but the edges will be left mostly untouched.
There are some exceptions to this ‘rule’ (see this post) but generally preformed tumbling media is superior to grit media for deburring, deflashing and edge rounding.