I recently had to grease up a ton of gears inside a compact model.
After trying fingers, a paintbrush and a screwdriver, I began to get frustrated trying to get the stuff on the gears and off me!
I eventually cut a small strip of brake shim copper. It is very thin and flexible.
I clamped it into the jaws of a craft knife where the blade usually goes and it works well to get to the hard to reach spots.
Being flexible yet stiff, it pushes the grease in exactly where you want it as you turn the gears.