There isn't a built in way to do this with Sprytile, but it is possible to do this. How you approach it would depend on where you intend to use the asset.
If it's for sketchfab or a blender render the easiest way would be to create multiple tiles with the animation frames, and then use bone animations to hide the unwanted frames somewhere in the model that is hidden from view.
If it's for a game model, the most efficient way to do this is to animate the UVs in the engine. This would require some extra setup in separating the animated parts of the mesh and creating the system that animates the UVs in the engine which is a bit out of the scope for a forum post.