The continuous development of synchrotron-based experimental techniques in the X-ray range provides new possibilities to probe the structure and the dynamics of bulk materials down to inter-atomic distances. However, the interaction of intense X-ray beams with matter can also induce changes in the structure and dynamics of materials. A reversible and non-destructive beam induced dynamics has recently been observed in X-ray photon correlation spectroscopy experiments in some oxide glasses at sufficiently low absorbed doses, and is here investigated in a (Li$_2$O)$_{0.5}$(B$_2$O$_3$)$_{0.5}$ glass. The characteristic time of this induced dynamics is inversely proportional to the intensity of the X-ray beam, with a coefficient that depends on the chemical composition and local structure of the probed glass, making it a potentially new tool to investigate fundamental properties of a large class of disordered systems. While the exact mechanisms behind this phenomenon are yet to be elucidated, we report here on the measurement of the exchanged wave-vector (and thus length-scale) dependence of the characteristic time of this induced dynamics, and show that it follows the same power-law observed in vitreous silica. This supports the idea that a unique explanation for this effect in different oxide glasses should be looked for.