  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 Brolin 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.
