Design space exploration (DSE) is a key activity in embedded design processes, where a mapping between applications and platforms that meets the process design requirements must be found. Finding such mappings is very challenging due to the complexity of modern embedded platforms and applications. DSE tools aid in this challenge by potentially covering sections of the design space that could be unintuitive to designers, leading to more optimised designs. Despite this potential benefit, DSE tools remain relatively niche in the embedded industry. A significant obstacle hindering their wider adoption is integrating such tools into embedded design processes. We present two contributions that address this integration issue. First, we present the design space identification (DSI) approach for systematically constructing DSE solutions that are modular and tuneable. Modularity means that DSE solutions can be reused to construct other DSE solutions, while tuneability means that the most specific DSE solution is chosen for the target DSE problem. Moreover, DSI enables transparent cooperation between exploration algorithms. Second, we present IDeSyDe, an extensible DSE framework for DSE solutions based on DSI. IDeSyDe allows extensions to be developed in different programming languages in a manner compliant with the DSI approach. We showcase the relevance of these contributions through five different case studies. The case study evaluations showed that non-exploration DSI procedures create overheads, which are marginal compared to the exploration algorithms. Empirically, most evaluations average 2% of the total DSE request. More importantly, the case studies have shown that IDeSyDe indeed provides a modular and incremental framework for constructing DSE solutions. In particular, the last case study required minimal extensions over the previous case studies so that support for a new application type was added to IDeSyDe.