  We consider spin-flip dynamics of configurations in
$\{-1,1\}^{\mathbb{Z}^d}$, and study the time evolution of concentration
inequalities. For "weakly interacting" dynamics we show that the Gaussian
concentration bound is conserved in the course of time and it is satisfied by
the unique stationary Gibbs measure. Next we show that, for a general class of
translation-invariant spin-flip dynamics, it is impossible to evolve in finite
time from a low-temperature Gibbs state towards a measure satisfying the
Gaussian concentration bound. Finally, we consider the time evolution of the
weaker uniform variance bound, and show that this bound is conserved under a
general class of spin-flip dynamics.
