A local Langlands correspondence for n-dimensional l-adic algebraic families of Galois representations has recently been established, demonstrating the compatibility of local Langlands with congruences and deformations. This correspondence amounts to an isomorphism between a moduli space of integral l-adic Galois representations and the integral Bernstein variety for $GL_n$. In this talk we will discuss work in progress with Dat, Helm, and Kurinczuk, toward a generalization of this method to arbitrary reductive groups that split over a tamely ramified extension.