I would not obsess about warming up with a trombone. Or worry much about a euphonium. Certainly in the circumstances being described.
I do definitely warm the instrument up (and keep it relatively warm) by breathing into it regularly (but not in a way that ends up being obvious to the audience). Now a tuba is an order of magnitude different. In similar circumstance with a tuba, I'm breathing into the tuba (and through each of the valve circuits) on pretty much a continual basis as I'm sitting there.
And if you're in the common situation in such venues of playing with an out of tune piano or organ of some sort, or a choir that isn't singing in tune or is tuned to some odd pitch, then "tuning" is a concept that hardly applies. Every year I play in a small and particularly odd group at a Christmas "concert" and "sing-along" for one of the (let's say, not very high end) senior "living centers" in the area. The piano almost certainly hasn't been tuned in over 50 years and won't play a reasonable scale or chord of any sort. We try to avoid playing with a pianist, but sometimes don't succeed. We also have an eclectic group of musicians from middle school age to 70s, and with instruments that include an unpredictable mixture of high and low brass (mostly well played), flutes, maybe a guitar, a viola, at times a marching baritone, a couple of saxophones (who are very good), and at times some "handheld percussion". I know this is hard to visualize. But the audience enjoys it anyway.
You often have to work with what you've got, and compromises must be made. Just do your best to make your own part work.
Gary Merrill
Wessex EEb Bass tuba (DW 3XL or 2XL)
Mack Brass Compensating Euph (DE N106, Euph J, J9 euph)
Amati Oval Euph (DE 104, Euph J, J6 euph)
1924 Buescher 3-valve Eb tuba (with std US receiver), Kelly 25
Schiller American Heritage 7B clone bass trombone (DE LB K/K10/112/14 Lexan, Brass Ark MV50R)
1947 Olds "Standard" trombone (Olds #3)