For algebraic curves $C$ over a closed field, a correspondence on $C$ is a the same thing as a divisor, and so, a line bundle on $C \times C$. Can I assume that this simplification does not extend to the case of general varieties? Does there exist a nice characterization of those correspondences 'expressible' as vector bundles?

In the case of finite fields, is there a bundle formation of the Frobenius correspondence $(v,$Fr$(v)) \in V \times V$ in terms of bundles?

I am particularly interested in the case of projective and Grassmannian spaces.