My idea of "warming-up" has always been to:
1. get your brain "turned-on" (get focused on playing only and
2. get your air moving freely and easily and oxygenating the brain and body
3. get your facial muscles literally "warmed-up" to 102 degrees (Muscles work best at this temperature, I am told) with fresh blood flowing through more open capillaries
4. get your tongue working fluidly
5. get your fingers loosened -up and under control
6. get all the above elements working together in balance
Maybe some people have all the above turned on all the time....cool.... some people my take 3 notes to get in gear, others may take 20 minutes...
Jake said warm-up if you have time, but don't be a slave to it...you CAN play without warming-up if your mechanics are good and your head "where it oughta be."
I saw him do this on a studio gig in the late 70's....we went to one of the Dallas studios for a jingle session....he said he hadn't had the horn out of the case in a couple of weeks. He walked in, sat down, pulled the horn out played about 3 notes, and the recording light went on. He played great.
SO MUCH of what we do is individualized. I tell my students that playing a brass instrument, from picking a mouthpiece to practicing etc...is like underwear: you gotta wear what fits....what is comfortable, non-binding, not too loose, not too tight, the right "cut," ....they laugh...no, they giggle....but they understand.
When I was young I needed no warm-up. We thought a "warm-up" was playing a march straight through. Now, as I am beginning to get "aged," I need to spend some time on all 6 points above to be sure I don't hurt myself again.
It depends also on where we are in our MENTAL and physical development, either the first time around, or in my case, the fourth.
One of the trends in education that I am actually GLAD to see (so much is bogus, politicized, or ivory-tower presuppositionally WRONG!) is that we are attempt to see each kid's individual strengths, weaknesses, and learning styles.
And maybe that's why Wynton doesn't sound like Andre doesn't sound like Bud doesn't sound like Vizzutti.