  Models with a nonequilibrium wetting transition display a transition also in
finite systems. This is different from nonequilibrium phase transitions into an
absorbing state, where the stationary state is the absorbing one for any value
of the control parameter in a finite system. In this paper, we study what kind
of transition takes place in finite systems of nonequilibrium wetting models.
By solving exactly a microscopic model with three and four sites and performing
numerical simulations we show that the phase transition taking place in a
finite system is characterized by the average interface height performing a
random walk at criticality and does not discriminate between the bounded-KPZ
classes and the bounded-EW class. We also study the finite size scaling of the
bKPZ universality classes, showing that it presents peculiar features in
comparison with other universality classes of nonequilibrium phase transitions.
