The mechanisms of comprehension during language processing remains an open question. Classically, building the meaning of a linguistic utterance is said to be incremental, step-by-step, based on a compositional process. However, many different works have shown for a long time that non-compositional phenomena are also at work. It is therefore necessary to propose a framework bringing together both approaches. We present in this paper an approach based on Construction Grammars and completing this framework in order to account for these different mechanisms. We propose first a formal definition of this framework by completing the feature structure representation proposed in Sign-Based Construction Grammars. In a second step, we present a general representation of the meaning based on the interaction of constructions, frames and events. This framework opens the door to a processing mechanism for building the meaning based on the notion of activation evaluated in terms of similarity and unification. This new approach integrates features from distributional semantics into the constructionist framework, leading to what we call Distributional Construction Grammars.