Depends on how this satellite was created.
Like with Moon - it's more about that part of Earth crust was thrown into orbit, giving remaining plates free space to move. Normally planetary crust is more rigid and have way less activity since all plates are tightly coupled and there's there's nowhere to move. By removing one of places rest was given ability to shuffle around, moving back and forth.

Like Mars have two satellites, but since his crust is "unbroken" - nothing happening there.

> objects at the far edges of the universe are actually travelling at constant velocity from their reference

That's the neat trick - they are not traveling at all! Expansion of Universe is happening because space between everything is growing, not because something is flying away.

Closest analogue would be - draw two points on a balloon and start inflating it. Distance between this two points would start to increase without them moving by themselves.