  It has recently been shown that the Hawking evaporation process of
$(3+1)$-dimensional Schwarzschild black holes is characterized by the
dimensionless ratio $\eta\equiv\tau_{\text{gap}}/\tau_{\text{emission}}\gg1$,
where $\tau_{\text{gap}}$ is the characteristic time gap between the emissions
of successive Hawking quanta and $\tau_{\text{emission}}$ is the characteristic
timescale required for an individual Hawking quantum to be emitted from the
Schwarzschild black hole. This strong inequality implies that the Hawking
cascade of gravitons from a $(3+1)$-dimensional Schwarzschild black hole is
extremely {\it sparse}. In the present paper we explore the semi-classical
Hawking evaporation rates of {\it higher}-dimensional Schwarzschild black
holes. We find that the dimensionless ratio
$\eta(D)\equiv{{\tau_{\text{gap}}}/{\tau_{\text{emission}}}}$, which
characterizes the Hawking emission of gravitons from the $(D+1)$-dimensional
Schwarzschild black holes, is a {\it decreasing} function of the spacetime
dimension. In particular, we show that higher-dimensional Schwarzschild black
holes with $D\gtrsim 10$ are characterized by the relation $\eta(D)<1$. Our
results thus imply that, contrary to the $(3+1)$-dimensional case, the
characteristic Hawking cascades of gravitons from these higher-dimensional
black holes have a {\it continuous} character.
