The author presents an exhausting description of Lipschitz continuous and some other classes of nonlinear superposition operators acting in modular metric spaces of functions of a real variable of finite generalized variation in the sense of \textit{M. Schramm} [Trans. Am. Math. Soc. 287, 49--63 (1985; Zbl 0567.26009)] with values in metric semigroups and cones. A complete description of generators for Lipschitz continuous, bounded and some other classes of superposition Nemytskii operators mapping in these semigroups and cones is given, which extends recent results by \textit{J. Matkowski} and \textit{J. Miś} [Math. Nachr. 117, 155--159 (1984; Zbl 0566.47033)] and some other authors.