Gunsmithing can you put a muzzle brake ina tumbler to clean it?

Yeah you can. I just put mine it my stainless tumbler when I got my Ross break. Now I did it when it was still raw metal, so I didn't need to worry about finish. So that would be the only thing you may need to consider.

The reason I put mine in the tumbler was to take out any burrs and to smooth everything out. Also to note, if this is a new break. I would either Time it after you tumble it or only time it to about 11:30. I timed mine prior to tumbling at 11:30 and after it tumbled, it timed perfect at 12.
 
I'd be more inclined to soak it for a while in your preferred cleaner (assuming it is just carbon build-up/fouling we are talking about needing to clean) and then clean by hand per usual, or to use a sonic cleaner, than to use a vibratory tumbler like one used for brass.

That said...walnut or corn media won't cause your brake any harm aside from perhaps a slight change in the finish (i.e. - if your brake is bead blasted/dull finish then the tumbler could "polish" it somewhat).
 
I'd be surprised at just how much cleaning the walnut will do to baked on carbon residue, it's not known to be particularly effective at cleaning the inside of brass cases.

Tumbling a brake in the SS media pins works exceptionally well. When it comes out the surface is all polished up too.

ETA: If you're really hankering to clean it (and I wouldn't be personally, I don't see a benefit to it) then if you cover return shipping I'll put it through my tumbler for you and get it back to you in about 2 weeks.
 
A bit off the subject but I've used plain old vinegar to help loosen and remove lead and carbon deposits
from a SS compensator which was mounted on a Ruger MK II pistol. I dunked the whole thing into the vinegar and let it sit for a few days.
It will loosen the deposits but will still require some work with a brush and picks but it does help.
Note, it will attach the finish and give it a frosted, matt, type appearance. Do not use the vinegar with anything that has bluing!