So, in typing the title of this question I was recommended this awesome one, which confirmed my guess that this effect "propagates" at the speed of sound (though I just had a feeling, I don't really understand why). I also only have high-school level physics experience.
I feel like I don't know physics well enough to really know what to ask correctly, but I'll try to explain.
So, I rotate my arm, and, even though tendons are just pulling it at a certain location, the rest of it follows suit. Why is that? What is happening at an atomic/molecular scale that ends up "conveying forces" over a distance?
And, something else I don't understand- what is so special about the speed of sound that makes it this "fundamental unit of translation", or something? Maybe the better question is "what processes at an atomic/molecular scale lead to the speed of sound being associated with all of these behaviors", which ties both questions together.