I've had rust (the Lee stuff seems worse for it) on dies and lock rings and press handles, etc.
For the outside surfaces I use a little 0000 steel wool and light oil and then wipe it off with a soft cloth. The rubbing removes the worst of the rust and the left over oil helps protect the surfaces.
I'll admit, I seldom look at the inside of the dies. I have a few times, but more for looking for a need for wax removal (bullet lube) on the pistol bullet seating/crimping dies than anything else. I've always wondered (believed??) that the case lube on the non carbide dies helped protect the inside of the die surfaces from rust. I know I've used it for lube on the press moving parts several times over the years. It doesn't seem to run off and make a dirty mess in the floor like oil does (and I use oil sometimes).
Oh, my stuff is currently in an unheated/uncooled garage. It used to be in my "reloading shed" about 60 yds. from the house. It wasn't heated/cooled either but was more subject (it seems) to the weather.
I remember reading that some hunters in the Pacific Northwest use wax on their shotguns/rifles to help keep them from rusting. Something like that might work for outside die surfaces, lock rings, the press handles, etc.
