Let K be a number field, let ϕ(x) ∈ K(x) be a rational function of degree d > 1, and let α ∈ K be a wandering point such that ϕ n (α) = 0 for all n > 0. We prove that if the abc-conjecture holds for K, then for all but finitely many positive integers n, there is a prime p of K such that v p (ϕ n (α)) > 0 and vp(ϕ m (α)) 0 for all positive integers m < n. Under appropriate ramification hypotheses, we can replace the condition v p (ϕ n (α)) > 0 with the stronger condition vp(ϕ n (α)) = 1. We prove the same result unconditionally for function fields of characteristic 0 when ϕ is not isotrivial.