Finite mixture regression models are useful for modeling the relationship between response and predictors, arising from different subpopulations. In this article, we study high-dimensional predic- tors and high-dimensional response, and propose two procedures to deal with this issue. We propose to use the Lasso estimator to take into account the sparsity, and a penalty on the rank, to take into account the matrix structure. Then, we extend these procedures to the functional case, where predictors and responses are functions. For this purpose, we use a wavelet-based approach. Finally, for each situation, we provide algorithms, and apply and evaluate our methods both on simulations and real datasets.