Motivated by local-global compatibility in the $p$-adic Langlands program, Emerton and Helm (and others) studied how the local Langlands correspondence for $GL(n)$ can be interpolated in Zariski families. In this talk I will report on joint work with C. Johansson and J. Newton on the interpolation in rigid families. We take our rigid space to be an eigenvariety $Y$ for some definite unitary group $U(n)$ which parametrizes Hecke eigensystems appearing in certain spaces of $p$-adic modular forms. The space $Y$ comes endowed with a natural coherent sheaf $\mathcal{M}$. Our main result is that the dual fibers $\mathcal{M}_y'$ essentially interpolate the local Langlands correspondence at all points $y \in Y$. This make use of certain Bernstein center elements which appear in Scholze's proof of the local Langlands correspondence (and also in work of Chenevier). In the pre-talk I will talk about the local Langlands correspondence, primarily for $GL(2)$.