Recent measurements in tokamak plasmas provide clear evidence for rapid nondiffusive transport and non-Gaussian fluctuations, and have been widely interpreted in terms of the sandpile and self-organized criticality (SOC) paradigms. Many of the statistical physics inferences that can be drawn from observations of, for example, avalanching transport remain to be explored. This paper will show that the statistical characterization of both experimentally observed and simulated avalanching transport phenomena reveals several points of contact with existing stochastic process models that have seldom been deployed in a plasma physics context. It will be shown that statistical physics techniques developed to model clustering of events can be used to characterize microscopic fluctuations in both local density and flux, as well as the global transport properties to which they give rise. This provides a fresh interpretation for some of the key aspects of observed critical gradient-driven transport phenomenology in tokamaks. In particular it provides new evidence for scale-free correlations in the fluctuations which drive the transport, and quantifies their distribution in terms of few-parameter non-Gaussian models. The correlation properties of density fluctuations can be interpreted in terms of random walk models, whereas flux fluctuations cannot: instead they can be described by the discrete negative binomial distribution, which again indicates clustering. Some of the spatio–temporal correlations considered emulate multichannel measurements in tokamaks, and it is shown how these can be used to characterize the transport of naturally arising coherent structures.