In this paper, we study distributionally risk-receptive and distributionally robust (or risk-averse) multistage stochastic mixed-integer programs (denoted by DRR- and DRO-MSIPs). We present cutting plane-based and reformulation-based approaches for solving DRR- and DRO-MSIPs without and with decision-dependent uncertainty to optimality. We show that these approaches are finitely convergent with probability one. Furthermore, we introduce generalizations of DRR- and DRO-MSIPs by presenting multistage stochastic disjunctive programs and algorithms for solving them. These frameworks are useful for optimization problems under uncertainty where the focus is on analyzing outcomes based on multiple decision-makers' differing perspectives, such as interdiction problems that are attacker-defender games having non-cooperative players. To assess the performance of the algorithms for DRR- and DRO-MSIPs, we consider instances of distributionally ambiguous multistage maximum flow and facility location interdiction problems that are important in their own right. Based on our computational results, we observe that the cutting plane-based approaches are 2800% and 2410% (on average) faster than the reformulation-based approaches for the foregoing instances with distributional risk-aversion and risk-receptiveness, respectively. Additionally, we conducted out-of-sample tests to showcase the significance of the DRR framework in revealing network vulnerabilities and also in mitigating the impact of data corruption.