I don’t know much about brass. I do know that I found I could hold a note on a clarinet much longer than I could hold a note singing, without circle breathing (which I never learned). With actual resistance, it’s much easier to control your breathing. That’s why untrained singers often pinch off the sound, adding resistance.
What I do know about brass is that it’s a bad idea to play when you have a headache, which suggests to me that it requires more abdominal pressure. I’ve seen faces turn red on particularly high notes, which also suggests this. (Though I do know there’s some sort of cheat for playing high but shrill notes–though I don’t know how it works.) High notes on a clarinet don’t require nearly as much effort, though you do have to work on tuning and tone more the higher you go.
Oh, and puffing your cheeks out is essential to circle breathing, or so I’m told. What you do is use the air in your cheeks to power the instrument while you’re breathing in.