The synchronous dynamics of an array of excitable oscillators, coupled via a generic graph, is studied. Non-homogeneous perturbations can grow and destroy synchrony, via a self-consistent instability which is solely instigated by the intrinsic network dynamics. By acting on the characteristic time-scale of the network modulation, one can make the examined system to behave as its (partially) averaged analogue. This result is formally obtained by proving an extended version of the averaging theorem, which allows for partial averages to be carried out. As a byproduct of the analysis, oscillation death is reported to follow the onset of the network-driven instability.