If the parts require a called out radius than the tumbler idea may not work. Tumbling works wonders removing any high spots, but expecting that process to create uniform radii will be a crap shoot depending on the tolerances called out, if any.
Also, any sharp corners will see a doubling of the material removal as they will be subjected to abrasion from multiple sides. So if the 'corners of the box' need to be held to the same radius as the edges than tumbling won't work as desired.
Having said that, if the parts just need a softening of the edges, in other words the print has a call out that says "break all sharp edges" than tumbling should do the job. Generally, a vibratory deburrer is used for that process and can be tailored to match the desired softening of the edges by the aggresiveness of the media used.
You should be able to do the same thing in your tumbler by choosing a media that will accomplish the desired outcome of roundness versus time spent tumbling. And if you have the parts and the tumbler experimenting would be a good thing.
