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We show that, over a nonarchimedean local field, the rigid refined localLanglands correspondence and associated endoscopic character identities forconnected reductive $G$ follow if one only has them for all such $G$ withconnected center. The strategy is to construct a projective system of centralextensions and then take limits of the Langlands correspondences (andendoscopic data) of each group in the system. As an application, we prove theequivalence of the rigid refined local Langlands correspondence and itsanalogue for isocrystals, generalizing the work of [Kal18] in the $p$-adiccase.

Isocrystals and limits of rigid local Langlands correspondences