  In 6D orbifold compactifications of N=1 supersymmetric gauge theories, the
one-loop behaviour of the 4D {\it effective} gauge coupling and of its beta
function are carefully investigated for momentum scales k^2 near the
compactification scale(s). It is showed that as k^2 is crossing the
compactification scale(s), there exists a smooth transition ("crossover") to a
power-like behaviour of the 4D effective coupling, with a continuous one-loop
beta function. Contrary to the naive expectation, the power-like behaviour sets
in at momentum scales {\it smaller} than the compactification scale, to suggest
a {\it global} effect of the Kaluza-Klein modes even below this scale. We argue
that the experimental observation of such behaviour is not necessarily a
quantum effect of the compactification, as often thought, and is dominated by
classical effects. Special attention is paid to convergence issues near
compactification scale(s) and to the scheme dependence of the results.
