The temperature profiles of magnetically confined plasmas can display distinctive longlived internal pedestals, caused by self-created transport barriers. This raises theoretical physics issues, distinct from the plasma physics modelling challenge, which concern the class of transport processes, physical principles, and control parameters that can generate such phenomenology. Here, we show that such structures can arise naturally through avalanching transport in a sandpile model. A single control parameter, that governs the spatial range of each rapid critical-gradient-triggered redistribution event, determines the occurrence and regularity of these effects.