  We investigate the class of infinite distance-transitive digraphs $D$ of
finite out-valency. We show that if $D$ is a weakly descendant-homogeneous in
such a class then either (1) $D$ has property $Z$ and the reachability relation
is not universal; or (2) $D$ does not have property $Z$, the reachability
relation is universal and $D$ has infinite in-valency. Also, we show that
earlier results, proved in the context of highly-arc-transitive digraphs, hold
under the weaker condition of distance-transitivity. Finally, we give a
description of a subclass of the class distance-transitive weakly
descendant-homogeneous digraphs for which the reachability relation is not
universal.
