An avalanche or ‘‘sandpile’’ model is discussed that generalizes the original self-organized criticality avalanche model of Bak, Tang, and Wiesenfeld [Phys. Rev. Lett. 59 , 381 (1987)] to include spatially extended local redistribution. A single control parameter specifies the spatial extent of local redistribution when the critical gradient is exceeded: this has profound consequences for nonlocal avalanching transport and for the dynamical behavior of the system, which are insensitive to other details such as the initial conditions and fluctuations in fueling or the critical gradient. The model possesses essentially two regimes of behavior. If the scale of nonlocal transport is of the order of the system size, the system is in the vicinity of a fixed point; in consequence the statistics of energy dissipation and length of avalanches are power law, and the time evolution is irregular (‘‘intermittent’’). If this scale is significantly smaller than the system size, the time evolution is quasiregular and follows a limit cycle. The first of these regimes appears relevant to the earth’s magnetosphere, where bursty transport and large scale reconfiguration (substorms) are observed. In this case the avalanche statistics have been inferred from observations of patches of intensity in the aurora, which may map to energy dissipation events in the magnetotail. The second regime displays significant links to the observed confinement phenomenology of magnetic fusion plasmas, corresponding to a broader range of model parameter space. For example, there is correlation between sandpile profiles, stored energy, and edge steepening on the one hand, and the control parameter on the other.