  Recently, the first author showed that the giant in random undirected graphs
is `almost' local. This means that, under a necessary and sufficient condition,
the limiting proportion of vertices in the giant converges in probability to
the survival probability of the local limit. We extend this result to the
setting of random digraphs, where connectivity patterns are significantly more
subtle. For this, we identify the precise version of local convergence for
digraphs that is needed.
  We also determine bounds on the number of strongly connected components, and
calculate its asymptotics explicitly for locally tree-like digraphs, as well as
for other locally converging digraph sequences under the `almost-local'
condition for the strong giant. The fact that the number of strongly connected
components is {\em not} local once more exemplifies the delicate nature of
strong connectivity in random digraphs.
