In recent years, dynamical relativistic jet simulation techniques have progressed to a point where it is becoming possible to fully numerically resolve gamma-ray burst (GRB) blast-wave evolution across scales. However, the modeling of emission is currently lagging behind and limits our efforts to fully interpret the physics of GRBs. In this work we combine recent developments in moving-mesh relativistic dynamics with a local treatment of non-thermal emission in a new code: GAMMA. The code involves an arbitrary Lagrangian-Eulerian approach only in the dominant direction of fluid motion which avoids mesh entanglement and associated computational costs. Shock detection, particle injection and local calculation of their evolution including radiative cooling are done at runtime. Even though GAMMA has been designed with GRB physics applications in mind, it is modular such that new solvers and geometries can be implemented easily with a wide range of potential applications. In this paper, we demonstrate the validity of our approach and compute accurate broadband GRB afterglow radiation from early to late times. Our results show that the spectral cooling break shifts by a factor of ~ 40 compared to existing methods. Its temporal behavior also significantly changes from the previously calculated temporary steep increase after the jet break. Instead, we find that the cooling break does not shift with time between the relativistic and Newtonian asymptotes when computed from our local algorithm. GAMMA is publicly available at: https://github.com/eliotayache/GAMMA.