This is how I envision it. Relative timelines are a bit tricky to get your head around but once you start using them you'll get to understand their use.
Attached is a document that I have re-created based on your example above. It's not that far away from what you were doing. The main concept of relative timelines is that whenever you continue one, it will begin from the last position and time (and anything else) that a "relative" timeline stopped at and continue from there to the end of the new timeline.
In your example (and from what I can gather) you want the object underneath the buttons to move to the next "mouse-overed" button and animate both the object and the "gears". This takes a little thought to begin with as you 2 separate objects animating (in my view) the gears and the object. Even though you had them on the same timeline animation I split them into 2 separate animations. The document might better explain. Have a look and come back if you have any questions
mouseovertest-vDBear.zip (37.4 KB)