  In this article we study the behavior of Newton polygons along
$\mathbb{Z}_p$-towers of curves. Fix an ordinary curve $X$ over a finite field
$\mathbb{F}_q$ of characteristic $p$. By a $\mathbb{Z}_p$-tower $X_\infty/X$ we
mean a tower of covers $ \dots \to X_2 \to X_1 \to X$ with $\mathrm{Gal}(X_n/X)
\cong \mathbb{Z}/p^n\mathbb{Z}$. We show that if the ramification along the
tower is sufficiently moderate, then the slopes of the Newton polygon of $X_n$
are equidistributed in the interval $[0,1]$ as $n$ tends to $\infty$. Under a
stronger congruence assumption on the ramification invariants, we completely
determine the slopes of the Newton polygon of each curve. This is the first
result towards `regularity' in Newton polygon behavior for
$\mathbb{Z}_p$-towers over higher genus curves. We also obtain similar results
for $\mathbb{Z}_p$-towers twisted by a generic tame character.
