Summary: This paper is part of a research programme on the structure of the moduli space of Lorentzian geometries, a Lorentzian analogue of Gromov-Hausdorff theory based on the use of the Lorentz distance as basic kinematical variable. We first prove results aimed at a better understanding of the tools available in this framework, such as the relationship between notions of closeness used to define limit spaces, and the properties of the auxiliary ‘strong’ Riemannian metric defined on each Lorentz space. Then, we examine concepts motivated by applications to quantum gravity, namely causality of the limit spaces and compactness of classes of Lorentz spaces.