The transition efficiency of atomic Bragg diffraction as mirrors and beam splitters in Bragg atom interferometers plays an essential role in impacting the fringe contrast and measurement sensitivity. This can be attributed to the properties of atomic sources, Bragg pulse shapes, the pulse duration, and the relative position deviation of the atoms and Bragg pulses. Here, we investigate the effect of the atomic source's diffusion and velocity width on the efficiency of Bragg diffraction of the moving cold atomic cloud. The transfer efficiency of Bragg mirrors and beam splitters are numerically simulated and experimentally measured, which are well consistent in comparison. We quantify these effects of atomic diffusion and velocity width and precisely compute how Bragg pulses' efficiencies vary as functions of these parameters. Our results and methodology allow us to optimize the Bragg pulses at different atomic sources and will help in the design of large momentum transfer mirrors and beam splitters in atom interferometry experiments.