  In this paper, we use geometric tools to study the structure of asymptotic
expanders and show that a sequence of asymptotic expanders always admits a
"uniform exhaustion by expanders". It follows that asymptotic expanders cannot
be coarsely embedded into any $L^p$-space, and that asymptotic expanders can be
characterised in terms of their uniform Roe algebra. Moreover, we provide
uncountably many new counterexamples to the coarse Baum--Connes conjecture.
These appear to be the first counterexamples that are not directly constructed
by means of spectral gaps. Finally, we show that vertex-transitive asymptotic
expanders are actually expanders. In particular, this gives a $C^*$-algebraic
characterisation of expanders for vertex-transitive graphs.
