Why does elevator input move the turn coordinator ball in steep turns?
In a steep turn you are making power changes, pitch changes and also making constant corrections with aileron to hold the bank angle, without even realizing it.  Once established in the turn some airplanes require a bit of in-turn aileron to hold the bank angle, some hold the bank with neutral aileron, and some require top aileron to keep from overbanking.  The top and down aileron inputs are inducing constantly changing adverse yaw forces which come and go, changing torque effects from the engine with power adjustments are producing changing yaw forces, and gyro precession from the prop from pitch motions are inducing yaw forces which come and go, and there are bumps, and you start running into your own wake.

In other words, you are in a machine pulled along by a big torque producing gyro that is slithering and sliding around in a gas with half a dozen forces and moments interacting on it simultaneously.  With all the subtle forces and inputs happening during a turn, I don't think you can identify and act on a single phenomenon like that.   What you have to do is, well, just do whatever it takes to hold the bank angle, altitude and center the ball and don't overthink it.  

If you fly gliders, which use a yaw string that is more sensitive than a ball, even without the torque and gyro effects of an engine, the yaw string drifts this way and that while in a turn, seemingly independent of the aileron position some of the time. You generally make stabs of rudder in concert with aileron, but sometimes the yaw string seems to have a mind of its own and you just do what you have to do with your feet.