  We prove that Fargues-Scholze's semisimplified local Langlands correspondence
(for quasisplit groups) with $\overline{\mathbb{F}}_\ell$-coefficients is
compatible with Deligne and Kazhdan's philosophy of close fields. From this, we
deduce that the same holds with $\overline{\mathbb{Q}}_\ell$-coefficients after
restricting to wild inertia, addressing questions of Gan-Harris-Sawin and
Scholze. The proof involves constructing a moduli space of nonarchimedean local
fields and then extending Fargues-Scholze's work to this context.
