Attach a script to your Background node

We need a reference variable to the TextureRect to access:

the position property of the TextureRect

the Texture property of the TextureRect

We need variables to keep track of 

the starting X and Y positions.

So that we can reset the tile once it has moved a certain distance.

As the scene becomes ready:

Capture & store the current x position

Capture & store the current y position

Create a function to move the tiles:

Move the tiles (Diagonally):

LEFT & UP

The tiles are 64 pixels wide.

Once they have moved 64 px from their starting position

We move them back to their starting position.

With ONE tile it would look like this:

Call the move_tiles() function in _process to make it loop

If you want to load a different tile using code.

You can load a different texture.