A practical consequence of the breakup of a liquid jet by the pinch-off singularity is the redistribution of volume. To the extent that volume concentrates into drops in the streamwise direction, pinch-off can lead to coarsening. The fundamental redistribution of volume by surface tension can be understood in the absence of pinch-off, however. We pose a simple model for the coarsening of connected spherical-cap drops in the absence of pinch-off. Our study shows that many properties of this simple model hold true for a general class of coupled elements. A system of N drops with pinned contact lines is coupled through a network of conduits. The system coarsens in the sense that, as time progresses, the volume becomes increasingly localized and ends up primarily in a single 'winner' drop. Numerical simulations show that the identity of the winner can depend discontinuously on the initial condition and conduit network. This motivates a study of the corresponding N-dimensional dynamical system. An analysis of the system yields analytic expressions for the fixed points and their energy stability, which depend only on the characteristic pressure-volume response of each element. The dynamic stability is shown to be identical to the energy stability, thus characterizing the number of stable and unstable manifolds at each fixed point. To determine which of the stable fixed points will be the winner, separatrix manifolds of the attracting regions are found using a method which combines local information from the eigenvectors at fixed points with global information from invariant manifolds obtained from symmetry. This method is used to explain phenomena observed in the numerical simulations.