In a work of T. Saito, the action of algebraic correspondences on the etale
cohomology of varieties over local fields with semistable reduction is related
to correspondences on smaller strata via weight spectral sequences. We give an
intersection theoretic construction of these correspondences. Under a
finiteness condition this enables us to compute them without involving the
blow-ups of products, and prove their compatibility with compositions. These
features are essential for the application to Shimura varieties.