The electron temperature profile response in ohmically heated Tokamak Fusion Test Reactor (TFTR) discharges to temporally and spatially sharp edge cooling from laser ablation of trace impurities is studied. The profile response is found to exhibit transport behaviour which cannot be described by transport coefficients that depend solely on local parameters. Thus, the transport behaviour is `non-local'. This non-local behaviour manifests itself as a core electron temperature rise in response to localized edge cooling as well as transient modification of sawteeth. A phenomenological model demonstrating overlapping non-local and non-linear effects is presented in the context of some alternative transport theories. Carbon flakes falling from the TFTR wall into a low power deuterium-tritium supershot produce this same core electron temperature rise. This paper complements observations by Gentle et al. (Gentle, K.W., et al., Phys. Rev. Lett. 74 (1995) 3620)