  Background: The astrophysical $r$-process occurs in an explosive
astrophysical event under extremely neutron-rich conditions, leading to
(n,$\gamma$)-($\gamma$,n) equilibrium along isotopic chains which peaks around
neutron separation energies of a few MeV. Nuclei with larger $Z$ are usually
produced by $\beta^-$-decay, but under certain conditions also $\alpha$-induced
reactions may become relevant for the production of nuclei with $Z+2$.
  Purpose: The uncertainties of the reaction rates of these $\alpha$-induced
reactions are discussed within the statistical model. As an example,
$\alpha$-induced ($\alpha$,n) and $(\alpha$,$x$n) reaction cross sections for
the neutron-rich $^{86}$Se nucleus are studied in detail.
  Method: In a first step, the relevance of ($\alpha$,n) and $(\alpha$,$x$n)
reactions is analyzed. Next the uncertainties are determined from a variation
of the $\alpha$-nucleus potential which is the all-dominant parameter for the
astrophysical $Z \rightarrow Z+2$ reaction rate.
  Results: It is found that the $r$-process flow towards nuclei with larger $Z$
is essentially influenced only by the $\alpha$-nucleus potential whereas the
other ingredients of the statistical model play a very minor role. This finding
is based on the fact that the flow towards larger $Z$ depends on the sum over
all ($\alpha$,$x$n) cross sections which is practically identical to the total
$\alpha$-induced reaction cross section.
  Conclusions: $\alpha$-nucleus potentials play an important role under certain
$r$-process conditions because the flow towards larger $Z$ depends sensitively
on the total $\alpha$-induced reaction cross section. The uncertainty of the
reaction rate is about a factor of two to three at higher temperatures and
exceeds one order of magnitude at very low temperatures.
