My first guess would be some oily residue in a die. It might even be an oil left in the die from manufacturing or put in the die to keep it from rusting during shipping and storage. It takes only a very tiny amount of any kind of oil in a die to do cause this. Now and then I'll use some Ed's Red, Hoppes, or whatever else is handy to clean out a dirty die, and no matter how many dry patches or clean rags I use after, it always seem I get dirty marks on cases for a little while after.
To see if I'm right, hit your die with some brake cleaner. That'll clean it right out and won't leave anything behind. The die will be dry as a bone. Since it's a Lee die set, use non-chlorinated brake cleaner so you won't hurt the rubber O-rings.