We establish a local 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 arbitrary 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 μ f of f, under the no potentially good reduction condition. This partly answers a question posed by Favre and Rivera-Letelier. We also obtain an equidistribution on the averaged value distribution of the derivatives of the iterated polynomials.