  We analyze and compute the semiclassical stress-energy flux components, the
outflux $\langle T_{uu}\rangle$ and the influx $\langle T_{vv}\rangle$ ($u$ and
$v$ being the standard null Eddington coordinates), at the inner horizon (IH)
of a Reissner-Nordstr\"om black hole (BH) of mass $M$ and charge $Q$, in the
near-extremal domain in which $Q/M$ approaches $1$. We consider a
minimally-coupled massless quantum scalar field, in both Hartle-Hawking ($H$)
and Unruh ($U$) states, the latter corresponding to an evaporating BH. The
near-extremal domain lends itself to an analytical treatment which sheds light
on the behavior of various quantities on approaching extremality. We explore
the behavior of the three near-IH flux quantities $\langle T_{uu}^-\rangle^U$,
$\langle T_{vv}^-\rangle^U$, and $\langle T_{uu}^-\rangle^H=\langle
T_{vv}^-\rangle^H$, as a function of the small parameter
$\Delta\equiv\sqrt{1-(Q/M)^2}$ (where the superscript "$-$" refers to the IH
value). We find that in the near-extremal domain $\langle
T_{uu}^-\rangle^U\cong\langle T_{uu}^-\rangle^H=\langle T_{vv}^-\rangle^H$
behaves as $\propto\Delta^5$. In contrast, $\langle T_{vv}^-\rangle^U$ behaves
as $\propto\Delta^4$, and we calculate the prefactor analytically. It therefore
follows that the semiclassical fluxes at the IH neighborhood of an evaporating
near-extremal spherical charged BH are dominated by the influx $\langle
T_{vv}\rangle^U$. In passing, we also find an analytical expression for the
transmission coefficient outside a Reissner-Nordstr\"om BH to leading order in
small frequencies (which turns out to be a crucial ingredient of our
near-extremal analysis). Furthermore, we explicitly obtain the near-extremal
Hawking-evaporation rate ($\propto\Delta^4$), with an analytical expression for
the prefactor (obtained here for the first time to the best of our knowledge).
[Abridged]
