  We undertake a systematic study of the approximation properties of the
topological and measurable versions of the coarse boundary groupoid associated
to a sequence of finite graphs of bounded degree. On the topological side, we
prove that asymptotic coarse embeddability of the graph sequence into a Hilbert
space is equivalent to the coarse boundary groupoid being topologically
a-T-menable, thus answering a question by Rufus Willett. On the
measure-theoretic side, we prove that measure-theoretic amenability resp.
a-T-menability of the coarse boundary groupoid are related to hyperfiniteness
and property almost-A resp. a version of "almost asymptotic embeddability into
Hilbert space". These results can be directly applied to spaces of graphs
coming from sofic approximations.
