  We study mean equicontinuous actions of locally compact $\sigma$-compact
amenable groups on compact metric spaces. In this setting, we establish the
equivalence of mean equicontinuity and topo-isomorphy to the maximal
equicontinuous factor and provide a characterization of mean equicontinuity of
an action via properties of its product. This characterization enables us to
show the equivalence of mean equicontinuity and the weaker notion of
Besicovitch-mean equicontinuity in fairly high generality, including actions of
abelian groups as well as minimal actions of general groups. In the minimal
case, we further conclude that mean equicontinuity is equivalent to discrete
spectrum with continuous eigenfunctions. Applications of our results yield a
new class of non-abelian mean equicontinuous examples as well as a
characterization of those extensions of mean equicontinuous actions which are
still mean equicontinuous.
