Let $X$ be an algebraic stack of finite type over a field $k$, and let $U\subset X$ be an open dense separated sub-scheme of $X$. Let $D=\text{Spec}~ k[[t]], D^*=\text{Spec}~ k((t))$. Fix a map $f:D^*\to U$. Consider the space of extensions $E_f$ of $f$ to a map $\overline{f}:D\to X$. It is easy to define it as a functor from affine schemes over $k$ to sets.
$\mathbf{Question}$: What condition on $X$ and $U$ would guarantee that $E_f(k)$ is finite when $k$ is a finite field? In fact, it would be nice to guarantee it geometrically: i.e. to show that $E_f$ is an algebraic space (or a scheme) of finite type over $k$ (when some condition on $X$ and $U$ is satisfied). For example, can it be sufficient that $U$ consists of closed points of $X$?
An example of such a good situation is this: take $X={\mathbb A}^2/{\mathbb G}_m$ where ${\mathbb G}_m$ on ${\mathbb A}^2$ by $\lambda: (x,y) \mapsto (\lambda x, \lambda^{-1}y)$; let also $U$ be ${\mathbb G}_m^2/{\mathbb G}_m$. On the other hand, the stack ${\mathbb A}^1/{\mathbb G}_m$ is bad from this point of view.
In some sense the above property means that $X$ is not too unseparated. So, basically, I am asking whether this property can be guaranteed by means of some explicit geometric condition on $X$ and $U$.
