I imagine the theoretical way would be to somehow link the shapes to a line and then bend the line itself so that there is no image distortion. I tried various bending methods but they all squish/expand the balls. I'm running out of ideas and I couldn't find any threads online that would cover this problem.
Create a null at the center of the circle implied by the finished logo. Shift-parent all the circles to that null (so they move to the null's location). With keyframes on their X position and rotation properties only, position them at each pose.