An a priori bound of rational functions on the Berkovich projective line

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.