Moving objects with their animations

Frequently I need to move tweened motion objects on my stage, so that the entire object with its motions are simply moved to a different position.  Whenever I select and drag objects, it always drags a pink path behind and interferes with the object's intended motions so the old position becomes a part of the motion of the object. How can I move objects around the stage without creating a pink motion path from the past position to the new position?

(BTW one thing I tried was to click-drag select that object with its whole motion, on the timeline itself. But when I do this, the object disappears because the act of selecting the frames has scrubbed the timeline to a position where that object is no longer in the scene. So, then the object is no longer visible on the stage and I cannot move it to its new position)

Thanks!
Laura