Consider the two roots (blue) of $f(z) = z^2 - e^{i\varphi}$ that "follow" $\varphi$ (black) with half speed (from this MSE-question):
Are there comparable statements of a similar kind: About the total or average "speed" of some "controlled" parameters in relation to the speed of some "controlling" parameters?
I know this is a very general and somehow vague question, but maybe someone bites.