Trumpet, I know. Let me tell you about trying to play taps, at the end of a ceremony, after freezing my ass ff in 30 degree weather. I think I kept my mouthpiece in my mouth constantly, so at least it was warm.
My trumpet teacher told me horror stories of horn players who had not done so. The mouthpiece freezes to your lips, and then when you try to pull it away….. um, did anyone ever lick anything metal when it was well below freezing? It’s not a pretty sight.
So my trumpet has been through all kinds of temperature changes, and after 30 or 40 years, it still sounds great. Maybe the tubes are worn very thin, that my repairperson says it could erode right through, but that hasn’t happened yet.
I don’t know flutes, though. I imagine the metal wouldn’t mind the temperature changes, and the expansion and contraction would be too little to make anything go out of true. The softer parts, as ladymerie suggests, might get worn a little faste.
Also, as loser says, if you play the instrument while cold, you will have some tuning issues. You’d think the horn would go sharp when cold (it gets smaller), but my horn always seems to go flat.