I load mine all six primer up just because of the gravity thing. Still training to get the reloads quick and consistent. Might switch if somebody can convince me of a better way.
Brass up, because if I reach for reloads I'd like to have them still in the carrier.
With the Mesa and TacStar type carriers, they can lose grip. Either the plastic body of the tacStar or the neoprene tube of the Mesa can become pliable enough to allow the rounds to fall out with recoil or running/movement. This usually occurs when the rounds are left in the carrier for long periods of time, the shotgun is stored in a hot vehicle, or something similar.
The elastic type carriers like the Vang and 3 Gun Gear types can usually hold brass down without any problems.
The brass down method is a little quicker, but just slightly so. Brass up requires a little more movement, but isn't that bad for speed.
There have also been schools that taught that you carried buck brass down towards the front and slugs brass up towards the rear.