Child Object won't move with parent

Hello all together! I have a problem which seemed to be simple at first but it's getting me stuck: I have a 2D character which moves based on a script (see below). I found a script to let the pupils follow the mouse courser but since I added this script to my character the pupils are fixed while the rest of the body moves. So I guess the problem lies within the script for the pupils. But I don't see the problem why this script blocks the pupils from moving. Do you guys have any ideas?

Here is the character hierarchy:

And the scripts:

Character Moving Script (Attached to 'Character')

 using UnityEngine;
 using System.Collections;
 
 public class CharacterWalking : MonoBehaviour {
 
     public float speed;             //Floating point variable to store the player's movement speed.
 
     private Rigidbody2D rb2d;       //Store a reference to the Rigidbody2D component required to use 2D Physics.
     Animator anim; 
 
     // Use this for initialization
     void Start()
     {
         //Get and store a reference to the Rigidbody2D component so that we can access it.
         rb2d = GetComponent<Rigidbody2D> ();
         anim = GetComponent<Animator> (); 
     }
 
     //FixedUpdate is called at a fixed interval and is independent of frame rate. Put physics code here.
     void FixedUpdate()
     {
         //Store the current horizontal input in the float moveHorizontal.
         float moveHorizontal = Input.GetAxis ("Horizontal");
 
         //Use the two store floats to create a new Vector2 variable movement.
         Vector2 movement = new Vector2 (moveHorizontal, 0);
 
         //Call the AddForce function of our Rigidbody2D rb2d supplying movement multiplied by speed to move our player.
         rb2d.AddForce (movement * speed);
     }
 
     void Update() {
         if (Input.GetKeyDown (KeyCode.A) || Input.GetKeyDown (KeyCode.D)) {
             anim.SetBool ("Walking", true);
         } 
 
         if (Input.GetKeyUp (KeyCode.A) || Input.GetKeyUp (KeyCode.D)) {
             anim.SetBool ("Walking", false);
         }
     }
 }
 

Pupil-Follow-Mouse script attached to 'Pupils'

 using UnityEngine;
 
 public class EyesFollow : MonoBehaviour {
 
     public Transform reference;
     public float factor = 0.25f;
     public float limit = 0.08f;
 
     private Vector3 center;
 
     void Start() {
         center = transform.position;
     }
 
     void Update () {
         Vector3 pos = Camera.main.ScreenToWorldPoint (Input.mousePosition);
         pos.z = 0.0f;
         Vector3 dir = (pos - reference.position) * factor;
         dir = Vector3.ClampMagnitude(dir, limit);
         Vector3 tt = center + dir;
         transform.position = center + dir;
     }
 }