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The allocation of disputed properties involves three levels in which fairness can be scrutinized: fairness in the initial allocation of rights on disputed properties, fairness in the transaction of allocated rights, and fairness of the end-state allocation. We explore the connections and implications of principles representing these three levels in a model of generalized exchange economies in which agents have preferences and claims over multiple goods, and the social endowment of each good may not be sufficient to satisfy all individual claims. To do so, we focus on rules obtained by combining a rights-assignment mechanism (to assign each profile of claims individual property rights over the disputed social endowment) and Walrasian exchange (or other individually rational exchange rules). We provide axiomatic characterizations of three canonical mechanisms (constrained equal awards, constrained equal losses, and proportional mechanisms), using variants of fairness based on the absence of envy. Our results are connected to focal contributions in political philosophy, as well as in the literature on fair allocation and rationing problems. They also provide rationale for popular schemes in environmental economics (such as markets for pollution permits).