  We give cases in which nearby cycles commutes with pushforward from sheaves
on the moduli stack of shtukas to a product of curves over a finite field. The
proof systematically uses the property that taking nearby cycles of Satake
sheaves on the Beilinson-Drinfeld Grassmannian with parahoric reduction is a
central functor together with a "Zorro's lemma" argument similar to that given
by Xue to prove the smoothness of cohomology sheaves at unramified places. As
an application, for automorphic forms at the parahoric level, we characterize
the image of tame inertia under the Langlands correspondence in terms of
two-sided cells.
