Extensions of the standard model with gauge symmetry $SU(3{)}_{c}\ensuremath{\bigotimes}SU(4{)}_{L}\ensuremath{\bigotimes}U(1{)}_{X}$ (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 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 flavor changing neutral current 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 ${K}^{+}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}\overline{\ensuremath{\nu}}\ensuremath{\nu}$, ${K}_{L}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}\ensuremath{\nu}\overline{\ensuremath{\nu}}$, ${K}_{L}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}{l}^{+}{l}^{\ensuremath{-}}$ ($l=e$, $\ensuremath{\mu}$) and ${B}_{d/s}\ensuremath{\rightarrow}{\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}}$.