  The set of differential equations obeyed by the redshift in the general
$\beta' \neq 0$ Szekeres spacetimes is derived. Transversal components of the
ray's momentum have to be taken into account, which leads to a set of 3 coupled
differential equations. It is shown that in a general Szekeres model, and in a
general Lema\^{\i}tre -- Tolman (L--T) model, generic light rays do not have
repeatable paths (RLPs): two rays sent from the same source at different times
to the same observer pass through different sequences of intermediate matter
particles. The only spacetimes in the Szekeres class in which {\em all} rays
are RLPs are the Friedmann models. Among the proper Szekeres models, RLPs exist
only in the axially symmetric subcases, and in each one the RLPs are the null
geodesics that intersect each $t =$ constant space on the symmetry axis. In the
special models with a 3-dimensional symmetry group (L--T among them), the only
RLPs are radial geodesics. This shows that RLPs are very special and in the
real Universe should not exist. We present several numerical examples which
suggest that the rate of change of positions of objects in the sky, for the
studied configuration, is $10^{-6} - 10^{-7}$ arc sec per year. With the
current accuracy of direction measurement, this drift would become observable
after approx. 10 years of monitoring. More precise future observations will be
able, in principle, to detect this effect, but there are basic problems with
determining the reference direction that does not change.
