  We report on the mechanism of energy transfer in van der Waals
heterostructures of the two-dimensional semiconductor WS$_2$ and graphene with
varying interlayer distances, achieved through spacer layers of hexagonal boron
nitride (hBN). We record photoluminescence and reflection spectra at interlayer
distances between 0.5 nm and 5.8 nm (0-16 hBN layers). We find that the energy
transfer is dominated by states outside the light cone, indicative of a
F\"orster transfer process, with an additional contribution from a Dexter
process at 0.5 nm interlayer distance. We find that the measured dependence of
the luminescence intensity on interlayer distances above 1 nm can be
quantitatively described using recently reported values of the F\"orster
transfer rates of thermalized charge carriers. At smaller interlayer distances,
the experimentally observed transfer rates exceed the predictions and
furthermore depend on excess energy as well as on excitation density. Since the
transfer probability of the F\"orster mechanism depends on the momentum of
electron-hole pairs, we conclude that at these distances, the transfer is
driven by non-thermalized charge carrier distributions.
