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Fire sales are among the major drivers of market instability in modern financial systems. Due to iterated distressed selling and the associated price impact, initial shocks to some institutions can be amplified dramatically through the network induced by portfolio overlaps. In this paper, we develop a mathematical framework that allows us to investigate central characteristics that drive or hinder the propagation of distress. We investigate single systems as well as ensembles of systems that are alike, where similarity is measured in terms of the empirical distribution of all defining properties of a system. This asymptotic approach ensures a great deal of robustness to statistical uncertainty and temporal fluctuations. A characterization of those systems that are resilient to small shocks emerges, and we provide criteria that regulators might exploit in order to assess the stability of a financial system. We illustrate the application of these criteria for some exemplary configurations in the context of capital requirements and test the applicability of our results for systems of moderate size by Monte Carlo simulations.