We examine the transport behaviour of non-interacting particles in a simple channel billiard, at equilibrium and in the presence of an external field. We observe a range of sub-diffusive, diffusive and super-diffusive transport behaviours. We find nonequilibrium transport that is inconsistent with the equilibrium behaviour, indicating that a linear regime does not exist for this system. However, we define a 'weak' linear regime that may lead to consistency between the equilibrium and nonequilibrium results. Despite the non-chaotic nature of the dynamics, we observe greater unpredictability (complexity) in the transport properties than is observed for chaotic systems. This observation seems at odds with existing complexity measures, motivating some new measures that attribute importance to the collective behaviour, rather than to the behaviour of individual system components. Such measures provide a new key for characterizing complexity in real-world phenomena.