  We study the effects of possible flavor-violating operators in theories with
the TeV scale quantum gravity, in which the ordinary matter is localized on a
3-brane embedded in the space with $N$ extra dimensions, whereas gravity
propagates in the bulk. These operators are scaled by the fundamental Planck
mass $M_{Pf}\sim$ TeV and must be suppressed by the gauge family symmetries. We
study suppression of the most dangerous and model-independent operators.
Several points emerge. First, we show that the Abelian symmetries can not do
the job and one has to invoke non-Abelian $U(2)_F$ (or $U(3)_F$) symmetries.
However, even in this case there emerge severe restrictions on the fermion
mixing pattern and the whole structure of the theory. In order not to be
immediately excluded by the well-known bounds, the horizontal gauge fields {\it
must} be the bulk modes, like gravitons. For the generic hierarchical breaking
pattern the four-fermion operators induced by the tree-level exchange of the
bulk gauge fields are unsuppressed for $N = 2$. For $N > 3$ the suppression
factor goes as a square of the largest $U(2)_F$-non-invariant Yukawa coupling,
which implies the lower bound $M_{Pf} > 10$ TeV or so from the $K^0 - \bar K^0$
system. Situation is different in the scenarios when flavor Higgs fields live
on a ($3 + N'$)-brane of lower dimensionality than the gauge fields. The
further suppression of gauge-mediated operators can be achieved by an explicit
construction: for instance, if $U(2)_F$ is broken by a vacuum expectation value
of the doublet, the troublesome operators are suppressed in the leading order,
due to custodial SO(4) symmetry of the Higgs-gauge quartic coupling.
