Within a metal, electrons can be treated as a gas; a kinetic model first developed by Drude ~1900.
Such a gas has a very low heat capacity (confirmed by experiment), and sheds kinetic energy quickly until the electron gas and the surrounding material reach equilibrium temperature.
Most of the shed energy is in the form of phonons (quantized sound). The moving ("free") electrons are not bound to individual atoms ... they are coupled to the bulk of the metal crystal, and move quite freely. These are your carrier electrons, which make up the conduction band.
This is most easily seen in far-from equilibrium systems as when an ultrafast laser pulse is absorbed by a thin metal film: the electrons, having a much smaller mass, are rapidly accelerated, and their thermal energy increases in less than a picosecond, resulting in a two-temperature system consisting of (1) the hot electron gas, and (2) the metallic ion cores. Depending upon the electron coupling constant for the metal ballistic (very hot) electrons may easily be emitted, or may be heavily suppressed.
You can find literature on this by searching Google Scholar for "electron two temperature model".
Note: I did my doctoral thesis on this topic, measuring the phonon distribution across different crystal directions. The cooling curve is the typical exponential thermal decay as the electron gas and the metal temperatures equilibrate. For a purely optical experiment one sees changes in optical reflectivity, which is a measure of the surface electron density.