Let $X/\mathbb{F}_q$ be a smooth affine curve over a finite field of characteristic $p > 2$. In this talk we discuss the $p$-adic variation of zeta functions $Z(X_n,s)$ in a pro-covering $X_\infty:\cdots \to X_1 \to X_0 = X$ with total Galois group $\mathbb{Z}_p$. For certain ``monodromy stable'' coverings over an ordinary curve $X$, we prove that the $q$-adic Newton slopes of $Z(X_n,s)/Z(X,s)$ approach a uniform distribution in the interval $[0,1]$, confirming a conjecture of Daqing Wan. We also prove a ``Riemann hypothesis'' for a family of Galois representations associated to $X_\infty/X$, analogous to the Riemann hypothesis for equicharacteristic $L$-series as posed by David Goss. This is joint work with Joe Kramer-Miller.