Given this simple windmill:

If a mechanical piston is added to the side:

The entire windmill gets pushed if the sails are not attached to the windmill (haven't right clicked):

But if you right click the bearing to start it spinning:

Then when you push it with the piston, the sails pop off:

Is there any way to fix this? Or, is there a different Kinetic Source that can be pushed by the piston while active?

Contraptions currently do not support cascaded movement, meaning you cannot move one contraption as part of another. The fact that the windmill bearing can be pushed while rotating was a bug and will be fixed in the upcoming patch.
You can try Encased Fans or Furnace Engines for a more portable kinetic source.

Thank you for the reply and help!