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 $gl(2)$.