We present the general relativistic framework that underpins a moving-mesh relativistic hydrodynamic code. To enhance the performance of this framework, we have developed a specialized tetrad formulation that optimizes the use of the HLLC Riemann solver in a moving mesh. Our research examines the ejecta that emerge from compact object mergers or collapses. By incorporating this new framework, we can trace the radial movement of the ejecta starting from the point where strong gravity is present. Our findings demonstrate that the code can evolve the ejecta structure from the central black hole scale with a primary mass of M to distances of one hundred thousand M and beyond. Additionally, we provide the analysis of investigating the microphysics of ejecta outflow from compact object simulations, along with its long-term electromagnetic observations.