A contiguity dichotomy for two sequences of product measures is proved under the assumption of component measures belonging to a dominated experiment which is differentiable. This generalizes Eagleson's (1981) result for Gaussian measures. The dichotomy result is then used to generalize and clarify the results of Shepp (1965) and Steele (1986) with regards to finite Fisher information and equivalence dichotomies between two product measures, one with a fixed component measure and the second with rigidly perturbed component measures.