We consider multistage stochastic mixed-integer programs M-SMIPs, and show that these problems are asymptotically convex. That is, we show that the expected cost-to-go functions in the problem converge to convex functions as the variability of the random parameters in the model increases. To derive this result, we first propose a novel adapted Gomory relaxation for deterministic mixed-integer value functions by selectively relaxing some of the non-negativity constraints of the basic variables. We show that the adapted Gomory relaxation is semi-periodic, and equal to the original mixed-integer value function on large parts of its domain. Then, we use these results to prove the desired asymptotic convexity of M-SMIPs. In contrast to earlier work on the two-stage case, our results also hold when only part of the right-hand side parameters are random, e.g., when some later-stage decisions are binary.