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Lafforgue and Genestier-Lafforgue have constructed the global and (semisimplified) local Langlands correspondences for arbitrary reductive groups over function fields. I will explain some recently established properties of these correspondences regarding base change functoriality: existence of transfers for mod p automorphic forms through p-cyclic base change in the global correspondence, and Tate cohomology realizes p-cyclic base change in the mod p local correspondence. The proofs are based on a combination of equivariant localization arguments (inspired by work of Treumann-Venkatesh) and the theory of parity sheaves (due to Juteau-Mautner-Williamson).��

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