Traditional multistage stochastic optimization assumes the underlying probability distribution is known. However, in practice, the probability distribution is often not known or cannot be accurately approximated. One way to address such distributional ambiguity is to use distributionally robust optimization (DRO), which minimizes the worst-case expected cost with respect to a set of probability distributions. In this talk, we study multistage convex DRO with a finite set of scenarios. We illustrate that not all but only some scenarios might have an effect on the optimal value, and we formally define this notion for multistage DRO. In particular, we investigate problems where the distributional ambiguity is modeled stagewise by the total variation distance on conditional probability distributions. We show the resulting problem is a multistage risk-averse optimization with nested coherent risk measures formed by a convex combination of the worst-case and conditional value-at-risk. We conduct perturbation analysis with respect to a collection of scenarios being excluded and propose easy-to-check sufficient conditions for effectiveness. We explore effectiveness of scenario paths as well as scenarios conditional on the history of the stochastic process. Computational experiments illustrate the results on finance and energy problems.