The canalizing properties of biological functions have been mainly studied in the context of Boolean modelling of gene regulatory networks. An important mathematical consequence of canalization is a low average sensitivity, which ensures in particular the expected robustness to noise. In certain situations, the Boolean description is too crude, and it may be necessary to consider functions involving more than two levels of expression. We investigate here the properties of nested canalization for these multivalued functions. We prove that the average sensitivity of nested canalizing multivalued functions is bounded above by a constant. In doing so, we introduce a generalization of nested canalizing multivalued functions, which we call weakly nested canalizing, for which this upper bound holds.