The global Gan-Gross-Prasad conjectures relate the non-vanishing of certain automorphic periods (integral over subgroups) to that of certain Rankin-Selberg L-functions at their center of symmetry. It also admits a refinement, due to Ichino-Ikeda, in the form of an exact identity connecting these two invariants. Following an approach initiated by Jacquet-Rallis of comparing two (relative) trace formulas, W. Zhang completed by recent work of Y. Liu, W. Zhang, X. Zhu and myself have established both of these conjectures for unitary groups and cuspidal representations that are "stable" i.e. whose quadratic base-change is again cuspidal. In this talk, I will explain how to deal with the remaining cuspidal representations, sometimes called "endoscopic", to which the conjectures apply through a fine spectral expansion of parts of the Jacquet-Rallis trace formula for general linear groups. This is joint work with Pierre-Henri Chaudouard and Michal Zydor.