Using Raycasts to change the X and Y position of a moving GameObject?

Hi, so first off, to give a little background, I'm a student and have been trying to teach myself Unity for the past year...and it's been going fairly well! However, a lot of what I've made so far are prototypes and now I'm working on my first somewhat original concept.

The idea is that you're controlling a spaceship as it continually goes down a set track (I'm using Cinemachine's Dolly Track with Cart to handle that aspect of the game). The most basic gameplay element is that while your spaceship is moving along the track, you can use WASD to shift your local x and y position by a certain value within a kind of 3x3 grid square. So you start in the middle of the center grid square, and depending on which key (or combination of keys with diagonals) you hit you would shift your position smoothly to the center of the grid square in that respective direction.

I was actually able to figure out how to perform this movement in a way, even if its not the cleanest code, by shooting Raycasts in a direction upon a key press, seeing if it hit the collider of a GameObject with the tag "MovementPoint" and then using the Lerp function to move the Player. Both the 9 GameObjects with the tags "MovementPoint" and the Player are a child of the Cinemachine Cart:

     [Header("Set in Inspector")]
     public float xySpeed = 20f;         // The player's speed when shifting.
     public float forwardSpeed = 50f;    // The player's speed when moving forward along the Z axis.
 
     [Header("Public References")]
     public CinemachineDollyCart dollyCart;
     public Transform cameraParent;
 
     [Header("Set Dynamically")]
     // The center in X and Y coordinates of the current grid square.
     private float centerX = 0;
     private float centerY = 20;
  
     [Header("Remain Constant")]
     private float sideLength = 15;      // The length of each side of a grid square.
   
     
     private void Start()
     {
         // The player's localPosition is set to the center of the middle grid square.
         transform.localPosition = new Vector3(centerX, centerY, 0);
     }
 
     private void Update()
     {
         // Call to the LocalMove() method which handles the basic player movement.
         LocalMove();
     }
     
     private void LocalMove()
     {
         RaycastHit hit;
 
         // If W is pressed.
         if (Input.GetKeyDown(KeyCode.W) && isLocked == false)
         {
             if (Physics.Raycast(transform.localPosition, transform.TransformDirection(Vector3.up), out hit, 15f))
             {
                 if (hit.collider.gameObject.CompareTag("MovementPoint"))
                 {
                     centerY += sideLength;
                 }
             }
         }
         // If S is pressed.
         if (Input.GetKeyDown(KeyCode.S) && isLocked == false)
         {
             if (Physics.Raycast(transform.localPosition, transform.TransformDirection(Vector3.down), out hit, 15f))
             {
                 if (hit.collider.gameObject.CompareTag("MovementPoint"))
                 {
                     centerY -= sideLength;
                 }
             }
         }
         // If A is pressed.
         if (Input.GetKeyDown(KeyCode.A) && isLocked == false)
         {
             if (Physics.Raycast(transform.localPosition, transform.TransformDirection(Vector3.left), out hit, 15f))
             {
                 if (hit.collider.gameObject.CompareTag("MovementPoint"))
                 {
                     centerX -= sideLength;
                 }
             }
         }
         // If D is pressed.
         if (Input.GetKeyDown(KeyCode.D) && isLocked == false)
         {
             if (Physics.Raycast(transform.localPosition, transform.TransformDirection(Vector3.right), out hit, 15f))
             {
                 if (hit.collider.gameObject.CompareTag("MovementPoint"))
                 {
                     centerX += sideLength;
                 }
             }
         }
         // Change the player's localPosition based on the newly set values.
         transform.localPosition = Vector3.Lerp(transform.localPosition, new Vector3(centerX, centerY, transform.localPosition.z), xySpeed * Time.deltaTime);
     }

However, while this works when the player is still...if I change the Cinemachine Cart's speed to anything higher than 0, while it appears like the colliders of the movement points are moving in the Scene View...I can't move the player at all unless I make the colliders super long, and even then it seems like they're not moving with their GameObjects. Is there any way to have Physics update the position of the colliders to match that of their GameObjects so that even if they're consistently moving they can be hit by raycasts? Or is there a better way to do this? Sorry for the giant block, this is my first post.