  Extensions of the standard model (SM) with gauge symmetry SU(3)_c X SU(4)_L X
U(1)_X (3-4-1 extensions) where anomaly cancellation takes place between the
fermion families (three-family models) predict the existence of two new heavy
neutral gauge bosons which transmit flavor changing neutral currents (FCNC) at
tree-level. In this work, in the context of a three-family 3-4-1 extension
which does not contain particles with exotic electric charges, we study the
constraints coming from neutral meson mixing on the parameters of the extension
associated to tree level FCNC effects. Taking into account experimental
measurements of observables related to K and B meson mixing and including new
CP-violating phases, we study the resulting bounds for angles and phases in the
mixing matrix for the down-quark sector, as well as the implications of these
bounds for the modifications in the amplitudes of the clean rare decays K^+ ->
\pi^+ \bar\nu \nu, K_{L} -> \pi^0 \nu \bar\nu, K_L -> \pi^0 l^+ l^- (l=e,\mu)
and B_{d/s} -> \mu^+ \mu^-.
