This paper analyzes previously-proposed dynamic models of content-sharing networks from the point of view of global stability. Our focus is on models that track populations of participating peers as a function of the download progress achieved, described in previous research by a Partial Differential Equation where this progress is a fluid variable. We use such a model to identify conditions on the rate allocation function that make the dynamics preserve a suitable ordering of the state. This enables the application of tools of monotone dynamical systems. This is formally done with finite-dimensional, ordinary differential equation model in which the content fraction index is discrete. Our results apply both to the case of homogeneous upload bandwidths in the participating population, as well as the heterogeneous case with multiple populations of each bandwidth class.