Refraction and reflection are fundamental light-bending effects occurring at media boundaries. While traditional studies have focused on stationary boundaries or potentials, moving potentials exhibit unique speed-dependent laws of refraction. Here, two types of moving potentials are constructed by either cascading two relatively tilted waveguide arrays or introducing a moving potential barrier within one tilted array to control discrete-light refraction and reflection. The approach leverages the speed-dependent scalar and vector gauge potentials induced by tilting the waveguide arrays. By tailoring the relative speeds via tilting of these arrays, full control is achieved over positive to negative refraction and even an invisible condition with zero refraction. For moving potentials, a transition from reflectionless to reflective behavior is demonstrated by tailoring the relative moving of the potential barrier and tilted array, and also identify a speed-based total internal reflection (TIR) condition that is useful for light guiding purposes. These findings open avenues for precise on-chip beam control in optical communications and signal processing.