You should be able to offset a texture and animate with the mapping node (in Red). Try the Location X & Y slider.
If that doesn’t work there would be some other problem.
i know that part. and what you just said is exactly how it should work. but it doesn’t. only the green marked area is able to do it.
Additional info: the texture is mapped on a material which is applied to a vertex group (i mean the object has two materials applied to parts of the object) this is about one of them.
Are you keying the sliders? By hovering over the appropriate slider and hitting the I key? If you look at Ja12’s image you can see the x location has keys. Indicated by the green highlight. I don’t see that or anything in your timeline to indicate that the change in value is actually keyed.