i use NIKAL uber hi-temp anti-seize grease. comes with a brush applicator.
permatex (auto store) anti-seize grease works just fine, too, applied with a finger. just slobber it on and any excess will get pushed out.
the important thing about any muzzleloader is the immediate care given to bp residue after the last shot of the day is taken. if there's a wait before the residue in the barrel/chamber for cleaning, that gives the goop a chance to take hold of the steel and do nasty things, as seen in the first two images of this thread. a sloppy oily wet patch down the tube, does the trick. better yet, leave the wet patched rod in the barrel. it also pays to carry a small spray bottle of moose milk (1:6 or so ratio of water soluble oil to plain water) and liberally spritz the entire lock. while some folks use NAPA water soluble oil, i prefer ballistol - it's good for the wood, too, so being sloppy with moose milk or straight ballistol is never a concern.
with immediate care given to last fired muzzleloaders, the real cleanup when back at the ranch goes amazingly fast ... and with just plain tepid tap water - there will be no need for chemicals or concoctions. the rust and corrosion on this thread's barrel and plug required overnight soaking in pb-blaster to loosen it all up and then i used a bronze brush with ballistol for the final cleaning.