Im trying to a child object follow the path of the parent in the path that the parent has already traveled...

Im trying to a child/children object(s) follow the path of the parent in the path that the parent has already traveled, if the parent stops moving i would like the child/children to stop as well. Once the parent starts moving I would like the child/children to continue the path of the parent like trying to make a centipede type object for example. Can anyone point me in the right direction to get me started on how to do that? Any help would be appreciated. Thanks again.