How do I create a moving background composed of a seamless texture?

I can work pretty well with seamless textures in both Photoshop and Illustrator, but I'm having a problem figuring out how to use them in After Effects.

What I'm Trying to Do: Create a slowly moving background that uses one of my seamless textures as its basis.

It feels like something I should be able to do, but when I perform searches on this I encounter people who have exclusively used shape layers to achieve a slightly different outcome. Example: found one where a background changes in a kaleidoscopic fashion, but that's not what I'm going for. I'd like to make a diagonally moving field of whatever texture I choose.

The only method I can imagine (with my admittedly limited AE experience) would involve creating a premade image (maybe at 4K) which I could then move from corner to corner. You know... this is easier to show. I want to do this.​ (tried to embed; wouldn't work)

Is this the right thinking, or am I missing something more subtle and elegant? I find that is often the case with all these tools.

Thanks for your attention.