We present a stochastic version of the Cucker-Smale flocking dynamics described by a system of N interacting particles. The velocity aligment of particles is purely discontinuous with unbounded and, in general, non-Lipschitz continuous interaction rates. Performing the mean-field limit as N → ∞ we identify the limiting process with a solution to a nonlinear martingale problem associated with a McKean-Vlasov stochastic equation with jumps. Moreover, we show uniqueness and stability for the kinetic equation by estimating its solutions in the total variation and Wasserstein distance. Finally, we prove uniqueness in law for the McKean-Vlasov equation, i.e. we establish propagation of chaos.