Let $\mathcal{M}$ be an algebraic stack, and let $X$ be its coarse moduli space (assume it exists as a scheme).

We know that $h_X(Spec(k))=\mathcal{M}(Spec(k))$ if $k$ is algebraically closed. Is there anything intelligent we can say about $h_X(U)$ for a general scheme $U$?

For example, can you come up with an algorithm for knowing what $h_X(U)$ is that would be considerably easier than constructing $X$?