When it's necessary to full length size cases, I find the easiest way to remove lube is with Birchwood Casey Brass Cartridge Cleaner. I lube the inside necks too and this cleaner removes all traces of lube, inside & out, gives the brass a nice sheen also. Three or four minutes in the solution, a thorough rinse and let 'em dry. I put mine outside in the summer sun for a few hours and find something else to do while they're drying. If you're seeing a grayish residue in the primer pockets after drying this indicates that the rinse wasn't thorough enough. I suppose a short time in the oven at low temp would dry the cases quickly.
I have convenient memory loss so I don't remember what I paid for this cleaner but Midsouth handles it in a concentrate, requiring 2 oz cleaner with 1 qt. water. In addition, you can use it a few times.
However, nothing beats straight wall cases & carbide dies or Lee Collet dies for bottleneck cases that don't require full length sizing.