Eyeball Material with subtle movements?

I’ve recently created a pretty simple yet effective eyeball material for my characters. In case you’re interested in recreating the material, here’s my setup:
The normal map is from Epic’s Paragon characters. The eyeball and iris textures are both full size 1024 images that I’ve made in Photoshop. The iris is then masked and put on top of the eyeball material, which gives me the very handy effect that I can adjust the iris size for each instance to match the character’s face proportions.

After reading how much stuff is actually possible through materials alone, I was wondering if it would be possible to somehow animate this material’s UVs slightly so it gives the eyes this natural “darting” effect without actually hooking them up in the skeleton and animating them with bones.

Does anyone have an idea how to achieve such an effect?

You can split the UV channel with component mask and take only the X output and add a scalar value (UvX + Value = horizontal movement, you can also use material parameter collection to animate the movement in bps).