Let $X$ be an algebaic stack over a scheme $S$, for any $S$-scheme $Y$ we can consider the groupoid $X(Y)$ of $Y$-points. Denote by $\pi_0(X(Y))$ the set of isomorphism classes of the groupoid.

Are there known results that tell us anything about the finitness of $\pi_0(X(Y))$?

To explain what I have in mind, if $X$ is a zero dimensional scheme of finite type over a field $k$, then $\pi_0(X(k))$ is $X(k)$ itself and it is finite.