Another example "on the other side" like Tony's (in fact really a stacky case of Tony's example) is de-equivariantization: given a $G$-space $X$ you can recover quasicoherent sheaves on $X$ from sheaves on $X/G$ (ie equivariant sheaves) as eigenobjects for the natural action of $Rep(G)=QC(BG)$ on $QC(X/G)$.

This appears in various more disguised forms in math -- for example look at the paper of Arkhipov-Gaitsgory on representations of the small quantum group where exactly this structure appears (similarly in work of Frenkel-Gaitsgory on representations at the critical level, don't remember precisely where right now). To quote the abstract: "We show that the category of $u_\ell$-modules is naturally equivalent to the category of $U_\ell$-modules, which have a {\it Hecke eigen-property} with respect to representations lifted by means of the quantum Frobenius map $U_\ell\ti U(\check g)$, where $g$ is the Langlands dual Lie algebra. "
(Of course these examples are not unrelated to geometric Langlands, but maybe slightly different contexts for your question.)

Another kind of example is given by monodromic sheaves --- namely, you might ask to weaken the condition of equivariance of a sheaf under a group action by twisting, so rather than getting an invariant (aka equivariant) object (eigenobject with trivial eigenvalue) you get a nontrivial eigenvalue. The most familiar case of this is writing twisted D-modules on flag varieties (in the context of Beilinson-Bernstein) -- these are D-modules on $X=G/N$ which are not strongly equivariant for the torus, so the pullback by action to the torus times $X$ is not product with a trivial local system, but rather eigenobject with egienvalue a local system on the torus.

In any case I would claim the "interesting examples" won't come from considering a single correspondence like you suggest, but rather some monoidal category acting (eg via correspondences) on sheaves on your space. That monoidal category can be symmetric as in the Hecke examples, or a "group algebra" as in the monodromic example, or something else, and now you can ask for a kind of twisted invariance (objects that transform according to a given representation of your monoidal category). So I would rather ask, what are interesting examples of monoidal categories acting
on categories of sheaves, besides the (wealth of) examples given by group actions on varieties?