Context. Gamma-ray binaries are systems that radiate the dominant part of their non-thermal emission in the gamma-ray band. In a wind-driven scenario, these binaries are thought to consist of a pulsar orbiting a massive star, accelerating particles in the shock arising in the wind collision. Aims. We develop a comprehensive numerical model for the non-thermal emission of shock-accelerated particles including the dynamical effects of fluid instabilities and orbital motion. We demonstrate the model on a generic binary system. Methods. The model was built on a dedicated three-dimensional particle transport simulation for the accelerated particles that were dynamically coupled to a simultaneous relativistic hydrodynamic simulation of the wind interaction. In a post-processing step, a leptonic emission model involving synchrotron and inverse-Compton emission was evaluated based on resulting particle distributions and fluid solutions, consistently accounting for relativistic boosting and γγ-absorption in the stellar radiation field. The model was implemented as an extension to the CRONOS code. Results. In the generic binary, the wind interaction leads to the formation of an extended, asymmetric wind-collision region distorted by the effects of orbital motion, mixing, and turbulence. This gives rise to strong shocks terminating the pulsar wind and secondary shocks in the turbulent fluid flow. With our approach it is possible for the first time to consistently account for the dynamical shock structure in particle transport processes, which yields a complex distribution of accelerated particles. The predicted emission extends over a broad energy range, with significant orbital modulation in all bands.