When I grease the FEL, I do pump the grit out. Using Mobil 1 red grease helps to notice the new grease oozing. With that said it does make a mess.
"My tools of the trade is a half roll of paper towels and big handful of Qtips."
I dont use any cleaning solvents, didnt think i needed any and a slight unnoticable film of grease is a rust preventative barrier.
First, pump grease in all the joints
Second, I have a piece if paper towel on the floor. Clean all the joints with the Qtips first, while throwing them in a pile on that paper towel. I can roll the Qtip and pick up the glop while not creating a bigger mess, using bith sides.
Third, wipe the rest of the joints with a piece of paper towel.
This is similar for my gun cleaning, so I always have a box of Qtips in the shop for the detail cleaning.