HI, all... I think that one should try to use similar (if not identical) rims on your mouthpieces. This way your embrochure doesn't have to get used to having different sized rims place on it.
Now, CUPS are a COMPLETELY different issue. In MY opinion, one should use the deepest cup that one can (no deeper than you can!) but each of the horns under consideration requires a different depth. The Flugelhorn should have the deepest cup, with the cornet a close second. The Trumpet should have a slightly shallower cup than the cornet, and the piccolo trumpet, a slightly shallower still cup.
For MY use, I use the mouthpiece that came with my Mirafone rotary flugel. It is quite deep and probably about a Schilke 18ish rim size (a Bach 1.5ish rim). On Bb cornet, I play a Wick 2 or 2B which also is a Bach 1.5ish sized rim. On rotary trumpet, I play a Schilke 22ish sized rim with a very deep cup, more like that of a flugelhorn, but on piston trumpet, I often play a GR mouthpiece custom made for me, but has a 22ish rim with a moderately deep cup (Bach B like). Finally, on picc and Eb soprano cornet, I play a Warburton 1D cup (around the Bach 1/Schilke 20 rim size) with a 12* backbore.
Thus, you see my rim size is pretty much in the Schilke 20 (Bach 1) range plus or minus a couple size units. It's the cup that differs most radically from instrument to instrument.
Of course, the SOUND that you get is most affected by the sound in your HEAD. I am far more agressive in my trumpet playing than my cornet or picc playing (which isn't to say that I'm particularly agressive there, either, but I try to "finesse" my playing even more on picc and cornet than I do on trumpet. (seduce your picc and cornet, don't try to force yourself on it!) Flugel, too, needs a light touch, at least for the kind of playing I do (symphonic and lyrical playing)
That's MY opinion, and worth every penny!