I'm sure it's more CYA than anything else -- but the fact that tumbler manufacturers care more about not being sued than whether you blow up your gun or not doesn't make the warning wrong. I do admit that the only "empirical" evidence I've run across for over-pressure due to powder break-down are the anecdotal "I know I guy who knows a guy who..." type that I tend to ignore.
However, I also know that smokeless powder granules will fracture under friction and that subjecting material like that to a lot of collisions (as is going to happen in tumbling) will cause it to break down somewhat. I also know that, other things being equal, powder in flake form will generate gas faster than in rod or ball form (fracture is much more likely in large particles, and they have more mass in collision), and that finer particles will generate gas more quickly due to more surface area. This supports the idea that tumbling powder could cause it to generate finer particles, and potentially more pressure. Now, obviously, this is going to have a lot of variables -- I expect it would be less of a concern for ball powder than rod, and even less for flake. It would also be less a concern in cartridges that are full or nearly full (so the mean path between collisions is very small), etc.
In general, when safety is a concern, you should assume worst case, so I don't tumble live rounds. And I won't until someone explains to me the flaw in the model described above.