Objects move along a smoothed path, but their speed varies
Bezier curves are used for graphics, but they're not so good for motion as their first derivative is not constant.
You'd want a spline with a constant first derivative respect to the parameter (time). There are a jillion different kinds of splines out there. I was looking for a constant first derivative spline to suggest, then I remembered that ease-in and ease-out would be nice, and such a spline would prevent that.
So my suggestion is to reparametrize the curve, and in the process you might be able to add ease in/out. This document explains how.
It's probably difficult.