Controlling Airflow

You can’t play well without controlling your air. Brass instruments run on focused airflow — it’s what creates the buzz in the first place. How high, low, loud, soft, and expressively you can play, how long you can sustain a phrase, and how the instrument actually sounds all come down to your air control. Get this right and it’ll transform your range.
Breathing in
Try this: empty your lungs completely, then just wait — don’t force a breath. Open your mouth and let the air flow back in on its own. Breathe in a relaxed way, filling your lungs fully, and avoid a hasty, tense gasp. Fill your lungs from the bottom up, using your diaphragm/core: breathe in deeply letting your stomach push out, then fill the top of your chest too without lifting your shoulders — stay relaxed and let your chest expand in every direction. Keep your arms away from your body so your chest has room to expand.
Tip: top up your air at every chance you get — it’s free, so use plenty of it. Fill your lungs completely right before playing a note or phrase. A fully stretched chest naturally wants to recoil, and that recoil gives your air stream extra energy, letting you play powerfully.
Breathing out
Keep the air focused on the way out, and maintain that energy and focus right until the air runs out — don’t try to keep playing on empty. Don’t hold your breath either; breathe in right before you need to exhale, not well ahead of it.
Tip: more airflow gives you more volume; faster air speed raises pitch. Push the air out to create real pressure behind it, and bear down harder with your core as notes climb higher — more pressure gives you faster air.
Try this: keep a pinwheel spinning with a steady, focused stream of air. That same feeling is what carries you through a smooth legato passage.
For a masterclass in air control, listen to trombone great Dave Steinmeyer play — pay attention to how rarely he actually breathes throughout the piece.