  Let $G$ be a second-countable, locally compact group. In this article we
study amenable $G$-actions on Kirchberg algebras that admit an approximately
central embedding of a canonical quasi-free action on the Cuntz algebra
$\mathcal{O}_\infty$. If $G$ is discrete, this coincides with the class of
amenable and outer $G$-actions on Kirchberg algebras. We show that the
resulting $G$-C*-dynamical systems are classified by equivariant Kasparov
theory up to cocycle conjugacy. This is the first classification theory of its
kind applicable to actions of arbitrary locally compact groups. Among various
applications, our main result solves a conjecture of Izumi for actions of
discrete amenable torsion-free groups, and recovers the main results of recent
work by Izumi-Matui for actions of poly-$\mathbb{Z}$ groups.
