We study multistage distributionally robust linear optimization, where the uncertainty set is a ball of distributions defined through the nested distance (Pflug and Pichler 2012) centered at a scenario tree. This choice of uncertainty set, as opposed to alternatives like the Wasserstein distance between stochastic processes, takes account of information evolution, making it hedge against a plausible family of data processes. Our contributions are two-fold. First, we develop a recursive reformulation to evaluate the worst-case risk of a given policy and relate it to the conditional risk mapping with single-period Wasserstein distance. Second, under the stagewise independence assumption, we derive dynamic programming reformulations for finding the optimal robust policy and identify tractable cases when the uncertainty appears on the objective or the right-hand side.