I fear there won't be heat penalties. I can't prove it or anything, just a suspicion. I've seen fire arrows, cannons and muskets being used in the pouring rain so I fear that weather will have little to no effect on the battles.

I hope I'm wrong and that the video with the muskets, fire arrows and cannons in the rain was just for promotion or in an arcade setting. Those complexities were what made the battles fun and diverse.