stear clear of grease in a loaded bore, IMO; especially if it is going to stay loaded for a while. You will get powder contamination, the severity of which largely depends on the grease used and the ambient temps. Variable Powder contamination=variable velocity=variable POI. Either use a softened wax (I use 3 : 1 : .15 Paraffin,Crisco,and diesel anti-gel) or spit to lube things. Having strips of greased cloth in and around your bag will make everything nasty, and it'll pick up dirt/grit that you'll put down your bore. Juggling a rifle, strip of cloth, a knife, and a bottle/tin of lube is a recipe for dropping something or forgetting something.
As far as patch boxes are concerned, I'm sure some people kept loading supplies in there for the gun (a common thing for especially a small game gun is to keep a couple balls, a couple patches, and an extra flint in there, so you can just grab your rifle, and a pocket flask and go get a couple rabbits for dinner (don't need or want to be encumbered with a belly box or small shot pouch). Just enough stuff for a couple shots. Other people have kept spare nipples, nipple wrench/turnscrew, jags+worms+ball screws in there (if they have multiple calibers, so that the cleaning supplies stay with the rifle they're meant for). A multi compartment patch box helps to keep things organized, but are relatively uncommon today (easier to router the whole thing out lol).
When I used cloth patched round ball (I mostly use paper now), I kept precut patches (squares, 1.5x the length and width of the caliber) in a little pocket on the outside of my shot pouch, just under the flap. first thing after shooting, I grab a patch out, stuff it in my mouth, then measure and pour the powder and by the time I put that away and grab a ball, the patch is soaked up enough. Put that on the muzzle and put the ball on top and shove it down the barrel, the fouling will soak up a lot of the moisture in the patch (cleaning while you load, I suppose). For me, the water is just there to help clean the previous fouling out of the path of the ball, and get the ball+patch down the barrel (my first shot, I don't usually wet the patch even).
I used to do the whole strip thing, and cut each patch at the muzzle, but getting a consistent patch is difficult (to say the least). Plus I hated having it dangle around in the dirt and stuff (since I put it in my mouth lol). That's why I went to precut patches. They don't "freeze to the muzzle" in the winter (at least not in OH lol), your mouth is 98.6'F, even if the barrel is 32'F, it still takes a sec to drop the temp to freezing (considering saliva freezes at a lower temp than plain water). Center patch, ball, ram, retrieve rammer- that fast (I only use a short starter if the bore is heavily fouled, not something that is going to happen in a hunting situation... I hope lol). That being said, all my rifles are shallow grooved, and I don't need or want a hermetic fitting ball and patch to fill the grooves).
You don't need grease or oil for cloth-patched RB, just like you don't need "BP solvent"; some people want to sell you something, some people buy into it.