You really don't have to be concerned about carrying a flintlock in a special way, presuming you have a good lock well maintained, sharp flint, etc.. I carry mine, several different ones, every whichaway. In the crook of my elbow with the muzzle high, right or left, over my shoulder, down at my side on the balance point, muzzle or butt first, in the crook of one arm, muzzle down, etc., etc. I also don't worry about it on stand. I hunt on the ground and am always laying my guns on my outstretched legs or on the ground, lock up or down, no problem. I tend to prime with smaller amounts, so my prime can shift with gun position, that is no issue. I just never think about this question because I figured out a long time ago it's not a problem.
I think many people don't realize just how well a good flint lock works. In 1997-8 Peter Alexander did a survey of locks which involved some pretty demanding tests. All the locks were fired upside down and most passed. All locks were primed, the frizzen closed, a finger put over the opening on the barrel side of the lock, it was dipped under water for 1 full minute, excess water was blown off and the lock fired upside down. Again, almost all the locks passed.
I shoot a double-barrel flintlock shotgun, and if you think about it, many times when you do something 'good' for one lock it's 'bad' for the other. I ignore all such ideas, carry it however I want, snap it up and shoot in whatever position I have to, never a problem.
Spence