The long-standing problem of nonlocal energy transfer observed in a number of fast heating/cooling experiments in tokamaks and stellarators is discussed. The most representative and still unexplained are the results from cold-pulse experiments that demonstrate a rapid transient increase in the electron temperature in the plasma core in response to an abrupt cooling of the edge. This occurs on a timescale much faster than the diffusive one, which is incompatible with conventional local transport models. This paper contains a review of the most enigmatical observations and theoretical efforts to describe them, and offers an alternative theoretical approach to the problem. Its essential element is the incorporation of the induction effects and fast energy exchange between the plasma and the magnetic field into the traditional transport model. This makes an emphasis on the magnetic field as the main carrier of the interaction, which naturally explains the fast and nonlocal character of the process. The approach uses the ideas recently applied to the analysis of a closely related problem of 'missing power'. Accordingly, the flux-conserving evolution of the plasma equilibrium and evaluation of the integral energy balance are parts of the analysis.