The basic idea is that every property can be animated.
Also the technique is basic : move the timelineslider to a desired tick, go into Animation Mode, and modify the property in the Texture panel, disable Animation Mode.
In this sample Grid Texture.3dm (94.0 KB) I used “Offset X” and “Offset Y” to ‘shift’ the grid along the plane.
It’s a bit strange that the names of the properties in the Keyframe Editor don’t fully match those in the Texture panel (e.g. XYZ for in stead of UV)! Any idea why @marika_almgren?
PS The animation of the Grid doesn’t Preview. You have to step through to see the result.