This topic seems to resurface every few months. . . .
I have tried a number of different greases, and I finally shied away from all of them, because you have only two choices - leave the grease on the gun, or take it off after you shoot. If you leave it on, it WILL collect grit, and the grit WILL increase the wear on the bearing parts (the ones that rub each other). Taking the grease off every time you shoot is hard to do - it is hard to get it out of the crevices, and some grease is so dark you can't see where it is or what it is carrying or covering up.
So I use a fairly heavy light-colored engine oil. I apply a few drops to the relevant places before I shoot, and I use a rag to wipe the gun completely dry after I finish shooting for the day. I find that oil is quite easy to get out of the gun with a rag - even from the small crevices. AND I can see the dirt, small specks of powder, etc. and ensure I get that stuff out too.
FWIW, the oil I use is Mobil 1, and I use 10W-40. But I don't think the brand or weight of oil is very important - virtually any good grade of oil will do the trick if you replace it every time you shoot. (It helps if it is light colored, hence the Mobil 1.) I have used less than a quart in two years, and that includes giving some to my friends (a little dab'll do ya).
I should add that putting oil on my gun before I shoot takes about one minute, and taking it off the gun after I shoot takes 2-3 minutes. It's not a big chore. I bough a package of cheap rags at Home Depot more than a year ago, and they have lasted quite well - occasionally my wife washes them, and they come out looking like cheap rags that have been washed. But they still wipe up the oil.
I might mention that the Perazzi people recommend using only oil, not grease.
If there is one place that it may be worth using grease, it's on your choke tubes. They stay in there for a fair amount of time, and if they are not well lubricated, they can freeze up. And it's easy to wipe a choke tube completely clean on the outside and not very hard to wipe the muzzle end of the barrel with a rag and get all the grease out of there, too.
I use a very high grade of Lubriplate (would you believe "food grade"), which is probably overkill, but maybe the BB's realize I'm a high-class guy when they go by my high-class grease and resolve to do a better job when they get to the target. Whatever . . . the high grade grease makes me feel better, even if the choke tubes and/or the BB's don't care.
Kolar specifically recommends using grease on their choke tubes.
Ted K.