  We generalize Renault's notion of measurewise amenability to actions of
second countable, Hausdorff, \'etale groupoids on separable $C^*$-algebras and
show that measurewise amenability characterizes nuclearity of the crossed
product whenever the $C^*$-algebra acted on is nuclear. In the more general
context of Fell bundles over second countable, Hausdorff, \'etale groupoids, we
introduce a version of Exel's approximation property. We prove that the
approximation property implies nuclearity of the cross-sectional algebra
whenever the unit bundle is nuclear. For Fell bundles associated to groupoid
actions, we show that the approximation property implies measurewise
amenability of the underlying action.
