Using panners and or rotators on the texture withing the material will set them in motion, adjusting the parameters will give you control on what it’s doing. Just be careful, I’m not sure you can use a rotator AND a panner on the same texture, I tried this and got odd results in which the layers were accelerated 10 fold.
But I had some reasonable success in throwing 3 of the same texture in and adding a panner to 2 and a rotator to 1, then add them together before joining to the base colour.