I always brought a bottle brush with most of the handle the handle cut off, and a small bottle of dishwashing liquid. The brush worked better to get the milk residue out of the valves and the bottom of the cup. I just filled a travel-size shampoo bottle with the dishwashing liquid; you can get 4-oz empty bottles for under $1 at beauty supply stores.
If you cut down a wire bottle brush, use small pliers to twist down the cut end of the wire, so you don't cut yourself while using it.
A trick I always used for in-between rinses was to take the valve out and drop it into the cup with a little liquid soap, then fill the cup halfway, close it, put my finger over the spout, and shake vigorously; then take off the top and rinse the whole thing out. The loose valve inside acted as an agitator, causing the soap to foam up nicely and get most of the milk dregs out. This kept the cups from getting rancid before I could get back to the room to clean them properly.