Moving a set of objects does not work

I am currently working on medical research room in virtual reality. I imported a model of a head and its slices into Unity.

I orded it like this in the hierachy:

-tissue(2)                                <--- Headmodel
-Slices
     -X                                   <---Slice (X-Axis)
          -Handle                         <---Handle (where to grab)
     -Y                                   <---Slice (Y-Axis)
          -Handle                        
     -Z                                    <---Slice (Z-Axis)
          -Handle

I created this simple code that moves the Coordinate system of the slices according to the headmodel:

using System.Collections;
using System.Collections.Generic;
using UnityEngine;

public class SlicesFolloeModel : MonoBehaviour {

    public GameObject headmodel;

	// Use this for initialization
	void Start () {
        transform.position = headmodel.transform.position;
	}
	
	// Update is called once per frame
	void Update () {
        transform.position = headmodel.transform.position;
        transform.eulerAngles = headmodel.transform.eulerAngles;
    }
}

However it doesn’t work properly. I want to mention that I made the Slices configurable joints so they move in only one direction when grabbed.
I made a video that explains my problem a bit further.

Your help would be really appreciated!

If I understand your intent correctly, it seems to me that the effect could achieved by making the slices the children of the head model. you can still allow them to be moved along their local axes, if you would like them to move independent of the head, like in the video.