I've added a character controller to a capsule and even though it has a rigidbody attached, it won't use gravity and fall to the floor. As soon as I disable it though, it does. I even tried making artificial gravity through scripting (the script uses flags to detect whether or not the player is touching the floor):

Vector3 moveDir = Vector3.zero;

void Update () {
    moveDir.y -= gravity * Time.deltaTime;
    var flags = controller.Move(moveDir * Time.deltaTime);
    grounded = ((flags & CollisionFlags.CollidedBelow) != 0);
}

('controller' being the CharacterController)

That didn't even work. Why is this happening?

Edit

Here's the whole movement script:

    public float speed = 10;
 public float rotationSpeed = 2;

 private bool canRun = true;
 public float runSpeed = 15;
 public bool running;

 private bool canCrawl = true;
 public float crawlSpeed = 5;
 public bool crawling;
 public float standUpTime = 3;
 public float crawlHeight;
 private float normalHeight;
 private Vector3 tempCent;

 private bool canJump = true;
 public float jumpHeight = 5;
 public bool jumping;

 public float gravity = 1;

 private Vector3 moveDir;
 private Vector3 rotDir;
 private float moveHor;

 private CharacterController controller;
 private PlayerStats playerStats;
 private Transform player;

 void Awake () {

 player = transform;
 controller = player.GetComponent<CharacterController>();
 playerStats = player.GetComponent<PlayerStats>();
 normalHeight = controller.height;
 tempCent.y = controller.center.y/2;
 }

 void Update () {

 bool isControllable = playerStats.isControllable;
 moveDir.y -= gravity * Time.deltaTime;

 if(!isControllable){
 Input.ResetInputAxes();
 }else{
 if(controller.isGrounded){
 moveDir = new Vector3(Input.GetAxis("Horizontal"), 0, Input.GetAxis("Vertical"));
 moveDir = player.TransformDirection(moveDir);
 moveDir *= speed;

 moveHor = Input.GetAxis("Horizontal");

 if(!crawling && !jumping){
 canRun = playerStats.canRun;
 running = playerStats.canRun;
 }
 canJump = playerStats.energy > 0;

 if(moveHor > 0){
 rotDir = new Vector3(0, 1, 0);
 }else if(moveHor < 0){
 rotDir = new Vector3(0, -1, 0);
 }else{
 rotDir = new Vector3(0, 0, 0);
 }

 if(Input.GetButton("Jump")){
 if(canJump){
 moveDir.y = jumpHeight;
 canRun = false;
 canCrawl = false;
 controller.height = normalHeight;
 controller.center = tempCent;
 }
 }

 if(Input.GetKey(KeyCode.LeftShift)){
 if(canRun){
 moveDir *= runSpeed;
 canCrawl = false;
 running = true;
 }
 if(crawling){
 animation.Play("GetUp");
 canRun = true;
 crawling = false;
 controller.height = normalHeight;
 controller.center = tempCent;
 } 
 }else{
 running = false;
 }

 if(Input.GetKey("c")){
 if(canCrawl){
 moveDir *= crawlSpeed;
 controller.height = crawlHeight;
 tempCent.y -= 0.25F;
 controller.center = tempCent;
 canRun = false;
 canJump = false;
 crawling = true;
 }
 }else{
 crawling = false;
 }
 controller.Move(moveDir * Time.deltaTime);
 controller.transform.Rotate(rotDir * Time.deltaTime, rotationSpeed);
 }
 moveDir.y -= gravity * Time.deltaTime;
 }
 }
}