  We investigate the local chromatic number of shift graphs and prove that it
is close to their chromatic number. This implies that the gap between the
directed local chromatic number of an oriented graph and the local chromatic
number of the underlying undirected graph can be arbitrarily large. We also
investigate the minimum possible directed local chromatic number of oriented
versions of ``topologically t-chromatic'' graphs. We show that this minimum for
large enough t-chromatic Schrijver graphs and t-chromatic generalized Mycielski
graphs of appropriate parameters is the upper integer part of t/4+1.
