Stating the bleedin' obvious, I know - the connection between us and the trumpet is the moutpiece. From my experience, it'a about 50:50 between eminent players / teachers who advocate playing pitches on the mouthpiece only and other equally eminent musicians who do not use this method of practising.
In Busking composed by H K Gruber's, written for Halkan Hardenberger, some of the music is written to be performed on the mouthpiece only - give it a listen yourselves.
Anyway, I'm a sad enough creature to enjoy practising with the moutpiece only, bearing in mind that, for me, it's a different feeling from playing the moutpiece in the leadpipe. On reflection, delving deep into any subject often throws up diametrically opposed views from equally eminent advocates. Trumpet playing for me involves a certain amount of experimentation in the search for improvement in delivering a musical message.
Hooked On The Other Side of The Bell www.bobreeves.com/blog/category/podcast/
While travelling by bus, I have been enthralled with the in-depth interviews with eminent trumpeters conducted by John Snell, who is the trumpet specialist at Bob Reeves Brass in the USA. Many of my trumpet heroes are interviewed about their lives as musicians and turn out to be generous in terms of advice and tips. I love Doc Severinsen talking about being the "Stairwell King" and giving his advice on how to conduct oneself in life. Studio musician, Jon Lewis, takes a great deal of time and trouble to give a valuable insight on various playing techniques.