Applying coherent X-rays by the method of atomic-scale X-ray Photon Correlation Spectroscopy results in beam-induced dynamics in a number of oxide glasses. Here these studies are extended to rubidium and caesium borates with varying alkali contents. While no cumulative beam damage is observed, the observed rate of structural rearrangements shows a linear relation to the dose rate. In agreement with the increasing glass transition temperature, the rate of dynamics at given dose rate decreases with increasing alkali content, while the shape of the decay of correlations becomes progressively stretched. This behavior is explained in terms of faster dynamics of the alkali positions compared to the borate network. Finally, the q-dependent behavior of the correlation decay rate implies the observed dynamics to proceed via small-scale atomic displacements subject to de Gennes narrowing.