Let your base S be Spec k[x] (say k is an algebraically closed field), let X be Spec k[x,y], and let G be <b>G</b><sub>a,S</sub>, with action over the point s given by g<sub>s</sub>(x<sub>s</sub>) = (sg<sub>s</sub>) + x<sub>s</sub>.  This action is transitive away from zero, so the orbit of the zero section is a plane with a slit.  This is not an open subscheme of a closed subscheme of X, because it is not a scheme.