  We establish a locally uniform a priori bound on the dynamics of a rational
function $f$ of degree $>1$ on the Berkovich projective line over an
algebraically closed field of any characteristic that is complete with respect
to a non-trivial and non-archimedean absolute value, and deduce an
equidistribution result for moving targets towards the equilibrium (or
canonical) measure $\mu_f$, under the no potentially good reductions condition.
This partly answers a question posed by Favre and Rivera-Letelier.
