  We investigate Ramsey expansions, the coherent extension property for partial
isometries (EPPA), and the existence of a stationary independence relation for
all classes of metrically homogeneous graphs from Cherlin's catalogue. We show
that, with the exception of tree-like graphs, all metric spaces in the
catalogue have precompact Ramsey expansions (or lifts) with the expansion
property. With two exceptions we can also characterise the existence of a
stationary independence relation and coherent EPPA.
  Our results are a contribution to Ne\v{s}et\v{r}il's classification programme
of Ramsey classes and can be seen as empirical evidence of the recent
convergence in techniques employed to establish the Ramsey property, the
expansion property, EPPA and the existence of a stationary independence
relation. At the heart of our proof is a canonical way of completing
edge-labelled graphs to metric spaces in Cherlin's classes. The existence of
such a ``completion algorithm'' then allows us to apply several strong results
in the areas that imply EPPA or the Ramsey property.
  The main results have numerous consequences for the automorphism groups of
the Fraisse limits of the classes. As corollaries, we prove amenability, unique
ergodicity, existence of universal minimal flows, ample generics, small index
property, 21-Bergman property and Serre's property (FA).
