We study the transfer of a resource from a group of suppliers to a group of demanders through links in a network. The analysis is relevant to situations where institutional constraints bar the use of the price mechanism: the allocation of workloads under fixed salaries, a commodity under disequilibrium prices, etc. In these contexts suppliers and demanders naturally have single-peaked preferences. We evaluate transfer rules on the basis of the “replacement principle” (Thomson, J Econ Theory 76(1):145–168 1997; Moulin, Q J Econ 102:769–783 1987), the requirement that a change in an agent’s preferences affects all other agents in the same direction in terms of welfare. We find that the only Pareto-efficient, participation-compatible, replication-invariant, and envy-free rule satisfying an appropriate formulation of the replacement principle is the “egalitarian rule” introduced by Bochet et al. (Theor Econ 7:395–423 2012).