  We formulate a trajectorial version of the relative entropy dissipation
identity for McKean$-$Vlasov diffusions, extending the results of the papers
[FJ16,KST20a], which apply to non-interacting diffusions. Our stochastic
analysis approach is based on time-reversal of diffusions and Lions'
differential calculus over Wasserstein space. It allows us to compute
explicitly the rate of relative entropy dissipation along every trajectory of
the underlying diffusion via the semimartingale decomposition of the
corresponding relative entropy process. As a first application, we obtain a new
interpretation of the gradient flow structure for the granular media equation,
generalizing the formulation in [KST20a] developed for the linear
Fokker$-$Planck equation. Secondly, we show how the trajectorial approach leads
to a new derivation of the HWBI inequality, which relates relative entropy (H),
Wasserstein distance (W), barycenter (B) and Fisher information (I).
