An interesting extension of the standard model is based on the electroweak gauge group $\mathrm{S}\mathrm{U}(3{)}_{L}\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)$. It requires three generations to cancel anomalies, treats the third generation differently than the first two, and has a rich phenomenology. There are several models, distinguished by the embedding of the charge operator into the $\mathrm{S}\mathrm{U}(3{)}_{L}$ group and by the choice of fermion representations. In this brief report, we consider flavor-changing neutral currents in these models, concentrating on the $P\ensuremath{-}\overline{P}$ mass difference, where $P=(K,D,B,{B}_{s})$, as well as $B\ensuremath{\rightarrow}K{l}^{+}{l}^{\ensuremath{-}}$, $B\ensuremath{\rightarrow}{\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}}$, and ${B}_{s}\ensuremath{\rightarrow}{\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}}$ decays. Although the $P\ensuremath{-}\overline{P}$ mass difference has been considered previously in some models, the rare $B$ decays are new. We find that the strongest bounds come from the $B\ensuremath{-}\overline{B}$ and ${B}_{s}\ensuremath{-}{\overline{B}}_{s}$ mass difference.