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We prove It{\^o}'s formula for the flow of measures associated with a jumpprocess defined by a drift, an integral with respect to a Poisson randommeasure and with respect to the associated compensated Poisson random measure.We work in $\mathcal{P}_{\beta}(\mathbb{R}^d)$, the space of probabilitymeasures on $\mathbb{R}^d$ having a finite moment of order $\beta \in (0, 2]$.As an application, we exhibit the backward Kolmogorov partial differentialequation stated on $[0,T] \times \mathcal{P}_{\beta}(\mathbb{R}^d)$ associatedwith a McKean-Vlasov stochastic differential equation driven by a Poissonrandom measure. It describes the dynamics of the semigroup associated with theMcKean-Vlasov stochastic differential equation, under regularity assumptions onit. Finally, we use the semigroup and the backward Kolmogorov equation to provenew quantitative weak propagation of chaos results for a mean-field system ofinteracting Ornstein-Uhlenbeck processes driven by i.i.d. $\alpha$-stableprocesses with $\alpha \in (1,2)$.

It{\^o}'s formula for the flow of measures of Poisson stochastic integrals and applications