The global steady state of a system in thermal equilibrium exponentially favors configurations with lesser energy. This principle is a powerful explanation of self-organization because energy is a local property of a configuration. For nonequilibrium systems, there is no such property for which an analogous principle holds, hence no common explanation of the diverse forms of self-organization they exhibit. However, a flurry of recent work demonstrates that a local property of configurations called "rattling" predicts the global steady states of a broad class of nonequilibrium systems. We interpret this emerging physical theory in terms of Markov processes to generalize and prove its main claims. Surprisingly, we find that the idea at the core of rattling theory is so general as to apply to equilibrium and nonequilibrium systems alike. Its predictions hold to an extent determined by the relative variance of, and correlation between, the local and global "parts" of an arbitrary steady state. We show how these key quantities characterize the local-global relationships of random walks on random graphs, various spin-glass dynamics, and models of animal collective behavior.