I enjoyed reading some interesting thoughts from multi-instrumentalist and music educator David Summer, who doubles quite effectively on flute and trumpet (and a few other instruments). I’m quoting a few highlights below, but definitely read the whole thing here.
I have seen no ill effects on either the trumpet embouchure or flute embouchure from playing both the flute and trumpet. I have no trouble going from one instrument to the other. In performance, I sometimes switch instruments, going from trumpet to flute or flute to trumpet, in the middle of a piece. This presents no problem at all.
As a multi-instrumentalist you will likely find more opportunities for performance… often people are glad to find that I can play both flute and trumpet and are happy to have me utilize that ability.
Certain fundamental musical concepts apply when playing any wind instrument. These include, embouchure development, breathing, pitch, articulation (tonguing), ear training, range, tone, technique (digital dexterity) and flexibility.
I believe that you should play the instruments that interest you and not be concerned about how one wind instrument embouchure might affect another. If you select instruments on the basis of those that you truly enjoy playing you will be more likely to keep playing and enjoying the enormous satisfaction that comes from making music.
Well said.