  We compute the variances of sums in arithmetic progressions of arithmetic
functions associated with certain $L$-functions of degree two and higher in
$\mathbb{F}_q[t]$, in the limit as $q\to\infty$. This is achieved by
establishing appropriate equidistribution results for the associated Frobenius
conjugacy classes. The variances are thus related to matrix integrals, which
may be evaluated. Our results differ significantly from those that hold in the
case of degree-one $L$-functions (i.e. situations considered previously using
this approach). They correspond to expressions found recently in the number
field setting assuming a generalization of the pair-correlation conjecture. Our
calculations apply, for example, to elliptic curves defined over
$\mathbb{F}_q[t]$.
