In all the years I've reloaded I've never cleaned my dies. I surely hope all that case lube I use helps keeps them from getting rusty.
I do watch their grease overflow port/s and pick them open if I remember too check prior to use.
Although I have worn or scared the bores badly on a couple non-carbide pistol dies mounted on a old Lee progressive press. Dirty dropped to the ground Nickel & steel surplus case's I'm sure had allot to do with that predicament.
As far as what to use to clean dies with? I would certainly consider the suggestions others here comment. Hopefully I have something appropriate in the garage to use for the purpose. I don't know if a bore brush is appropate as they all have a twisted steel wire that might make contact with a ridge or wall.
I use a towel on a dowel. There is something about starting over day like it was a new world that causes me to avoid it. When the inside of my die gets 'that finish' I apply the the leaver policy; I leaver the way I founder. I am not the fan of including chemicals as an answer to every problem.
It's unlikely the brass bore brush would scratch the inside of the die on its own.
But it could if there were something abrasive in there.
Like Doyle said, spray clean it and wipe it down avoids that.
The towel is embeddable, meaning if there is something in the die that can scratch the inside of the die damage can be avoided when the abrasive dirt, grit or grime is embedded into the towel.
No idea how people dont clean them at all.. Maybe when people say "clean dies?" they at least brush them out casually. I am fairly new to reloading and am pretty sure I am overly tumbling/polishing and still have quite a bit of sh*T it mine
A little solvent on a patch .../ for decapping and resizing dies in station 1 ( I disassemble them - and clean them ) - but I only clean them after a few thousand rounds.
A dry Q Tip - ...in stations 2 thru 5 ( powder drop, bullet seating and final crimp )...is about all you need to clean up the dies.