  Let $G$ be a split semi-simple group over a global function field $K$. Given
a cuspidal automorphic representation $\Pi$ of $G$ satisfying a technical
hypothesis, we prove that for almost all primes $\ell$, there is a cyclic base
change lifting of $\Pi$ along any $\mathbb{Z}/\ell\mathbb{Z}$-extension of $K$.
Our proof does not rely on any trace formulas; instead it is based on
modularity lifting theorems, together with a Smith theory argument to obtain
base change for residual representations. As an application, we also prove that
for any split semisimple group $G$ over a local function field $F$, and almost
all primes $\ell$, any irreducible admissible representation of $G(F)$ admits a
base change along any $\mathbb{Z}/\ell\mathbb{Z}$-extension of $F$. Finally, we
characterize local base change more explicitly for a class of representations
called toral supercuspidal representations.
