We develop a framework for the stochastic thermodynamics of a probe coupled to a fluctuating medium with spatio-temporal correlations, described by a scalar field. For a Brownian particle dragged by a harmonic trap through a fluctuating Gaussian field, we show that near criticality (where the field displays long-range spatial correlations) the spatially-resolved average heat flux develops a dipolar structure, where heat is absorbed in front and dissipated behind the dragged particle. Moreover, a perturbative calculation reveals that the dissipated power displays three distinct dynamical regimes depending on the drag velocity.