Equation discovery methods enable modelers to combine domain-specific knowledge and system identification to construct models most suitable for a selected modeling task. The method described and evaluated in this paper can be used as a nonlinear system identification method for gray-box modeling. It consists of two interlaced parts of modeling that are computer-aided. The first performs computer-aided identification of a model structure composed of elements selected from user-specified domain-specific modeling knowledge, while the second part performs parameter estimation. In this paper, recent developments of the equation discovery method called process-based modeling, suited for nonlinear system identification, are elaborated and illustrated in two continuous-time case studies. The first case study illustrates the use of the process-based modeling on synthetic data while the second case-study evaluates process-based modeling on measured data for a standard system-identification benchmark. The experimental results clearly demonstrate the ability of process-based modeling to reconstruct both model structure and parameters from measured data.