Caustic methods have been proposed for wavefront design to enable light beams propagating along curved trajectories, namely accelerating beams. Here we elaborate the complete construction, remarkable characteristics, and hidden constraints of these methods. It is found that accelerating beams based on the caustic design have not only a well-known curved intensity distribution but also a linear phase distribution along the caustic proportional to the curved length, as if light field indeed moved along the caustic. Moreover, with this characteristic, further light-ray analyses are implemented to illustrate the constraints of caustic design in different cases. We expect our work will clarify some confusion on the effectiveness and applicability of caustic methods, and thus facilitate the design of accelerating beams for various applications.