We study the low energy spin excitations of zigzag graphene nanoribbons of varying width. We find their energy dispersion at small wave vector to be dominated by antiferromagnetic correlations between the ribbon's edges, in accrodance with previous calculations. We point out that spin wave lifetimes are very long due to the semi-conducting nature of the electrically neutral nanoribbons. However, application of very modest gate voltages cause a discontinuous transition to a regime of finite spin wave lifetime. By further increasing doping the ferromagnetic alignments along the edge become unstable against transverse spin fluctuations. This makes the experimental detection of ferromagnetism is this class of systems very delicate, and poses a difficult challenge to the possible uses of these nanoribbons as basis for spintronic devices.