Self-correspondences on curves

Suppose there are two non-constant separable morphisms from a curve $D$ to a smooth curve $C$, everything is defined over a field $k$. A self-correspondence on $C$ is the data consisting of the curve $D$ and these two morphisms. Consider an algebraic closure $K$ of the field $k$. A self-correspondence can be thought of as a multi-valued map from $C(K)$ to itself, defined by polynomials with coefficients in the original field. We will discuss finite subsets of the curve $C$ which remain fixed under the self-correspondence. Following the paper by J. Bellaïche, I will explain when there are infinitely many such subsets, and how many there can be in the opposite case.