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A previously developed phase-field model with finite interface dissipation for binary dual-phase alloys out of chemical equilibrium is generalized to a multi-component multi-phase model in the framework of the multi-phase-field formalism, allowing the description of multiple junctions with an arbitrary number of phases and components. In multiple junctions, each phase concentration is assigned to a dynamic equation to account for finite interface dissipation, and its formulation is proposed in two different models. The overall mass conservation between the phases of a multiple junction is used in model I, whereas the concentrations of each pair of phases have to be conserved during the transformations for model II. Both models demonstrate the decomposition of the nonlinear interactions between different phases into pairwise interaction of phases in multiple junctions. They converge to the same equilibrium state while in non-equilibrium states different predictions are given.