Systems of Hecke eigenvalues on subschemes of Shimura varieties

We show that the systems of Hecke eigenvalues that appear in the coherent cohomology with coefficients in automorphic line bundles of any mod $p$ abelian type compact Shimura variety at hyperspecial level are the same as those appearing in any Hecke-equivariant closed subscheme. We also prove analogous results for noncompact Shimura varieties or nonclosed subschemes, such as Ekedahl-Oort strata, length strata and central leaves.