Turns out this only happens if you've previously edited the path that the object travels along. Is this probably a bug or am I just messing up and doing something wrong? Thanks
It is hard to tell much from the video. Based on all the blue I see, it looks like you have motion tweens assigned for the duration of the timeline. You say that one dot should stop, so I'd recommend that you remove the tween for the duration of the timeline where it is supposed to be stopped. For the keyframe where it starts beoing stopped, right click it and choose Remove Tween.
It appears you are using the newer version of timeline tweening. I cannot help much with that. I would recommend using classic tweens instead... but that's just me.