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

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