Is it possible to make an object with the follow path constraint stay at the end of the path even if the path moves? I tried setting it up so that the object is parented to the path, but that didn't work. It still goes off in a random direction when the path moves.
I suspect that you are selecting both when you only should move the curve. Select curve only when moving. Otherwise you will get something called double transformation, essentially moving it twice getting all kinds of weirdness.