Isocrystals and limits of rigid local Langlands correspondences

We show that, over a nonarchimedean local field, the rigid refined local Langlands correspondence and associated endoscopic character identities for connected reductive $G$ follow if one only has them for all such $G$ with connected center. The strategy is to construct a projective system of central extensions and then take limits of the Langlands correspondences (and endoscopic data) of each group in the system. As an application, we prove the equivalence of the rigid refined local Langlands correspondence and its analogue for isocrystals, generalizing the work of [Kal18] in the $p$-adic case.