  We determine when the equidistribution property for possibly moving targets
holds for a rational function of degree more than one on the projective line
over an algebraically closed field of any characteristic and complete with
respect to a non-trivial absolute value. This characterization could be useful
in the positive characteristic case. Based on the variational argument, we give
a purely local proof of the adelic equidistribution theorem for possibly moving
targets, which is due to Favre and Rivera-Letelier, using a dynamical
Diophantine approximation theorem by Silverman and by Szpiro--Tucker. We also
give a proof of a general equidistribution theorem for possibly moving targets,
which is due to Lyubich in the archimedean case and due to Favre and
Rivera-Letelier for constant targets in the non-archimedean and any
characteristic case and for moving targets in the non-archimedean and 0
characteristic case.
