Quantum systems that have never interacted can become nonlocally correlated through a process called entanglement swapping. To characterize nonlocality in this context, we introduce local models where quantum systems that are initially uncorrelated are described by uncorrelated local variables. This additional assumption leads to stronger tests of nonlocality. We show, in particular, that an entangled pair generated through entanglement swapping will already violate a Bell-type inequality for visibilities as low as 50% under our assumption.