Excellent question, with regrettably few answers! :^)
I personally expect that we know less than 10% of the significant things that go on during warm-up. So take the following with many large grains of salt ...
The one that I do believe in is transformer magnetization, specifically parafeed outputs, just because we have some experience with them. The turn-on and/or turn-off transient currents will magnetize the core of such a transformer. The extent depends on several factors, but it is sometimes enough to be audible. Music itself is a good-enough demagnetizing signal; I can't recall hearing a case where 30 minutes or so is not enough to mostly restore performance. It's not really a temperature thing - you have to run some music through the transformer.
Transistors are much more sensitive to temperature, both gain and operating point will shift significantly as the devices reach operating temperature. This is usually counteracted by the use of heavy negative feedback, which introduces its own problems, making it pretty hard to be sure what change causes which audible audible effect. And there's no automatic guarantee that warmer is better - just that it's stable and repeatable.
There are probably similar things going on in the speaker driver damping material - that goop on the surround (or the exotic rubbbery stuff it's made of) for instance. The stuff is surely sensitive to temperature, and has hysteresis (takes a "set") when not used.
Well, those are the examples that come to mind right now.
