Consider finitely many nets of multivariate càdlàg stochastic processes. We show that the vectors consisting of the respective minimizing points converge in distribution to a random closed set. This set is given as a cartesian product with factors which are equal to the set of all minimizing points of stochastic processes occurring as functional limits of the respective nets. If these limit processes have almost surely exactly one minimizer, then the vectors converge classically in distribution to the vector of these minimizers.