This paper proposes a formal framework based on Tree Adjoining Grammar (TAG) that aims to incorporate central tenets of Construction Grammar while integrating mechanisms from a psycholinguistically motivated variant of TAG. Central ideas are (i) to give TAG-inspired tree representation to various constructions including schematic constructions like argument structure constructions, (ii) to link schematic constructions that are extensions of each other within a network of constructions, (iii) to make the derivation proceed incrementally, (iv) to allow the prediction of upcoming constructions during derivation and (v) to introduce the incremental extension of schematic constructions to larger ones via extension trees in a usage-based manner. The final point is the major novel contribution, which can be conceptualized as the on-the-fly traversal of the inheritance links in the network of constructions. Moreover, we present first experiments towards a parser implementation. We report preliminary results of extracting constructions from the Penn Treebank and automatically identifying constructions to be added during incremental parsing, based on a generative language model (GPT-2).