  We investigate systems of interacting bosonic particles confined within
slab-like boxes of size L^2 x Z with Z<<L, at their three-dimensional (3D) BEC
transition temperature T_c, and below T_c where they experience a quasi-2D
Berezinskii-Kosterlitz-Thouless transition (at T_BKT < T_c depending on the
thickness Z). The low-temperature phase below T_BKT shows quasi-long-range
order: the planar correlations decay algebraically as predicted by the 2D
spin-wave theory. This dimensional crossover, from a 3D behavior for T > T_c to
a quasi-2D critical behavior for T < T_BKT, can be described by a transverse
finite-size scaling limit in slab geometries. We also extend the discussion to
the off-equilibrium behavior arising from slow time variations of the
temperature across the BEC transition. Numerical evidence of the 3D->2D
dimensional crossover is presented for the Bose-Hubbard model defined in
anisotropic L^2 x Z lattices with Z<<L.
