Fluorescence microscopy reveals that the contents of many (membrane-free) nuclear "bodies" exchange rapidly with the soluble pool whilst the underlying structure persists; such observations await a satisfactory biophysical explanation. To shed light on this, we perform large-scale Brownian dynamics simulations of a chromatin fiber interacting with an ensemble of (multivalent) DNA-binding proteins; these proteins switch between two states -- active (binding) and inactive (non-binding). This system provides a model for any DNA-binding protein that can be modified post-translationally to change its affinity for DNA (e.g., like the phosphorylation of a transcription factor). Due to this out-of-equilibrium process, proteins spontaneously assemble into clusters of self-limiting size, as individual proteins in a cluster exchange with the soluble pool with kinetics like those seen in photo-bleaching experiments. This behavior contrasts sharply with that exhibited by "equilibrium", or non-switching, proteins that exist only in the binding state; when these bind to DNA non-specifically, they form clusters that grow indefinitely in size. Our results point to post-translational modification of chromatin-bridging proteins as a generic mechanism driving the self-assembly of highly dynamic, non-equilibrium, protein clusters with the properties of nuclear bodies. Such active modification also reshapes intra-chromatin contacts to give networks resembling those seen in topologically-associating domains, as switching markedly favors local (short-range) contacts over distant ones.