  We study the time evolution of the R\'enyi entanglement entropies following a
quantum quench in a two-dimensional (2D) free-fermion system. By employing
dimensional reduction, we effectively transform the 2D problem into decoupled
chains, a technique applicable when the system exhibits translational
invariance in one direction. Various initial configurations are examined,
revealing that the behavior of entanglement entropies can often be explained by
adapting the one-dimensional quasiparticle picture. However, intriguingly, for
specific initial states the entanglement entropy saturates to a finite value
without the reduced density matrix converging to a stationary state. We discuss
the conditions necessary for a stationary state to exist and delve into the
necessary modifications to the quasiparticle picture when such a state is
absent.
