Coupled populations of identical phase oscillators may give rise to heteroclinic cycles between invariant sets where populations show distinct frequencies. We consider an example of four coupled phase oscillator populations consisting of two oscillators each, such that there are two heteroclinic cycles forming a heteroclinic network. While such networks cannot be asymptotically stable, their local attraction properties can be quantified by stability indices. We illustrate how the indices of both cycles in the network can be calculated in terms of the coupling parameters between oscillator populations and are able to give some conclusions about the non-asymptotic stability of the network as a whole. Our results elucidate how oscillator coupling influences sequential transitions along a heteroclinic network. We discuss challenges in applying this technique to more general systems and highlight similarities/differences between our stability results and those for simple networks of comparable architecture such as the extensively studied Kirk-Silber heteroclinic network.