  We develop a theory for heat transport via electromagnetic waves inside
media, and use it to derive a spatially nonlocal thermal conductivity tensor,
in terms of the electromagnetic Green's function and potential, for any given
system. While typically negligible for optically dense bulk media, the
electromagnetic component of conductivity can be significant for optically
dilute media, and shows regimes of Fourier transport as well as unhindered
transport. Moreover, the electromagnetic contribution is relevant even for
dense media, when in presence of interfaces, as exemplified for the in-plane
conductivity of a nanosheet, which shows a variety of phenomena, including
absence of a Fourier regime.
