We study multistage distributionally robust mixed-integer programs under endogenous uncertainty, where the probability distribution of stage-wise uncertainty depends on decisions made in previous stages. We first consider ambiguity sets defined by decision-dependent bounds on the first and second moments of uncertain parameters, and by the mean and covariance matrix that exactly match decision-dependent empirical ones. For both cases, we show that the subproblem in each stage can be recast as a mixed-integer linear program. Then we extend the ambiguity set in Delage and Ye (2010) to the multistage decision-dependent setting, based on which we derive mixed-integer semidefinite programming reformulations of the subproblems and develop methods for attaining lower and upper bounds for the multistage formulation. We also approximate the subproblem reformulations with a series of mixed-integer linear programs. We deploy the Stochastic Dual Dynamic integer Programming approach to solve our models with risk-neutral or risk-averse objectives, and conduct numerical studies using facility-location instances under different demand uncertainty settings.