  We show that time dependent couplings may lead to nontrivial scaling
properties of the surface fluctuations of the asymptotic regime in
non-equilibrium kinetic roughening models . Three typical situations are
studied. In the case of a crossover between two different rough regimes, the
time-dependent coupling may result in anomalous scaling for scales above the
crossover length. In a different setting, for a crossover from a rough to
either a flat or damping regime, the time dependent crossover length may
conspire to produce a rough surface, despite the most relevant term tends to
flatten the surface. In addition, our analysis sheds light into an existing
debate in the problem of spontaneous imbibition, where time dependent couplings
naturally arise in theoretical models and experiments.
