Based on recent advances in Benders decomposition and two-stage stochastic integer programming we present a new generalized framework to generate Lagrangian cuts in multistage stochastic mixed-integer linear programming (MS-MILP). This framework can be incorporated into decomposition methods for MS-MILPs, such as the stochastic dual dynamic integer programming (SDDiP) algorithm. We show how different normalization techniques can be applied in order to generate cuts satisfying specific properties with respect to the convex hull of the epigraph of the value functions, e.g. having a maximum depth or being facet-defining. We provide computational results to evaluate the efficacy and performance of different normalizations in our new framework and compare them with existing techniques from the literature.