In some magnetically confined plasmas, an applied pulse of rapid edge cooling can trigger either a positive or negative excursion in the core electron temperature from its steady state value. We present a new model which captures the time evolution of the transient, non-diffusive local dynamics in the core plasma. We show quantitative agreement between this model and recent spatially localized measurements (Inagaki et al 2010 Plasma Phys. Control. Fusion 52 075002 ) of the local time-evolving temperature pulse in cold pulse propagation experiments in the Large Helical Device.