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We give cases in which nearby cycles commutes with pushforward from sheaveson the moduli stack of shtukas to a product of curves over a finite field. Theproof systematically uses the property that taking nearby cycles of Satakesheaves on the Beilinson-Drinfeld Grassmannian with parahoric reduction is acentral functor together with a "Zorro's lemma" argument similar to that givenby Xue to prove the smoothness of cohomology sheaves at unramified places. Asan application, for automorphic forms at the parahoric level, we characterizethe image of tame inertia under the Langlands correspondence in terms oftwo-sided cells.

Unipotent nearby cycles and the cohomology of shtukas