Summary: Let \(f\) be a rational function of degree \(d>1\) on the projective line over a possibly non-archimedean algebraically closed field. A well-known process initiated by H. Brolin [Ark. Mat. 6, 103–144 (1965; Zbl 0127.03401)] considers the pullbacks of points under iterates of \(f\), and produces an important equilibrium measure. We define the asymptotic Fekete property of pullbacks of points, which means that they mirror the equilibrium measure appropriately. As application, we obtain an error estimate of equidistribution of pullbacks of points for \(C^1\)-test functions in terms of the proximity of wandering critical orbits to the initial points, and show that the order is \(O(\sqrt{kd^{-k}})\) upto a specific exceptional set of capacity 0 of initial points, which is contained in the set of superattracting periodic points and the omega-limit set of wandering critical points from the Julia set or the presingular domains of \(f\). As an application in arithmetic dynamics, together with a dynamical Diophantine approximation, these estimates recover Favre and Rivera-Letelier’s quantitative equidistribution in a purely local manner [C. Favre and J. Rivera-Letelier, Math. Ann. 335, No. 2, 311–361 (2006; Zbl 1175.11029)].