  We present analytical results for the Euclidean 2-point correlator of the
flavor-singlet vector current evolved by the gradient flow at next-to-leading
order ($O(g^2)$) in perturbatively massless QCD-like theories. We show that the
evolved 2-point correlator requires multiplicative renormalization, in contrast
to the nonevolved case, and confirm, in agreement with other results in the
literature, that such renormalization ought to be identified with a universal
renormalization of the evolved elementary fermion field in all evolved
fermion-bilinear currents, whereas the gauge coupling renormalizes as usual. We
explicitly derive the asymptotic solution of the Callan-Symanzik equation for
the connected 2-point correlators of these evolved currents in the limit of
small gradient-flow time $\sqrt{t}$, at fixed separation $|x-y|$. Incidentally,
this computation determines the leading coefficient of the operator-product
expansion (OPE) in the small $t$ limit for the evolved currents in terms of
their local nonevolved counterpart. Our computation also implies that, in the
evolved case, conservation of the vector current, hence transversality of the
corresponding 2-point correlator, is no longer related to the
nonrenormalization, in contrast to the nonevolved case. Indeed, for small flow
time the evolved vector current is conserved up to $O(t)$ softly violating
effects, despite its $t$-dependent nonvanishing anomalous dimension.
