Berkeley FrameNet is a lexico-semantic resource for English based on the theory of frame semantics. It has been exploited in a range of natural language processing applications and has inspired the development of framenets for many languages. We present a methodological approach to the extraction and generation of a computational multilingual FrameNet-based grammar and lexicon. The approach leverages FrameNet-annotated corpora to automatically extract a set of cross-lingual semantico-syntactic valence patterns. Based on data from Berkeley FrameNet and Swedish FrameNet, the proposed approach has been implemented in Grammatical Framework (GF), a categorial grammar formalism specialized for multilingual grammars. The implementation of the grammar and lexicon is supported by the design of FrameNet, providing a frame semantic abstraction layer, an interlingual semantic application programming interface (API), over the interlingual syntactic API already provided by GF Resource Grammar Library. The evaluation of the acquired grammar and lexicon shows the feasibility of the approach. Additionally, we illustrate how the FrameNet-based grammar and lexicon are exploited in two distinct multilingual controlled natural language applications. The produced resources are available under an open source license.