When holding very long notes, I usually run out of oxygen before I've emptied my lungs (air gets stale as they say). On horns that don't take much air, like soprano, I often have to vent through the corners of my mouth in order to maintain a good breathing cadence. But I'm a big guy with big lungs.
I've never really given breathing much thought. If I've got an unusually long phrase coming up, I'll take an extra deep breath. Otherwise I just take normal, quick breaths. Regardless, I think the key here is to maintain pressure with the abs rather than just rely on a relaxing diaphragm. I think this happens automatically for most people. But if you have a weak tone, you definitely need to take a good, deep breath and push with the abdomen.
Obviously circular breathing is impossible if you breath through the mouth, so we're talking about two distinct techniques here. But if you don't or can't circular breathe, breathing through the nose won't allow you to take big or quick enough breaths. So I would not recommend nose breathing outside of circular breathing.