$\DeclareMathOperator\Spec{Spec}$Let $k$ be a field. Let $S= \mathbb{A}^1_k = \Spec(k[x])$ and $X= \mathbb{A}^2_k = \Spec(k[x,y])$.The constant additive group $\mathbb{G}_a\times S$ over $S$ acts on $X$ by the equation $$ t \cdot (x,y) = (x, y+xt).$$ The orbit of the $0$ section $\phi: S \to X$ defined by $$ \phi(x) = (x,0)$$ consists of the the union $V \cup (0,0)$, where $V$ is the open complement of the $y$-axis ($x=0$), and $(0,0) \in \mathbb{A}^2$ is the origin. This can't possibly be flat over $S$, since the fiber over $x=0$ is $0$-dimensional while the fiber over every other point is $1$-dimensional.
I don't think that this image is locally closed either.
