  We study the non-equilibrium response function $R_{ij}(t,t')$, namely the
variation of the local magnetization $\langle S_i(t)\rangle$ on site $i$ at
time $t$ as an effect of a perturbation applied at the earlier time $t'$ on
site $j$, in a class of solvable spin models characterized by the vanishing of
the so-called {\it asymmetry}. This class encompasses both systems brought out
of equilibrium by the variation of a thermodynamic control parameter, as after
a temperature quench, or intrinsically out of equilibrium models with violation
of detailed balance. The one-dimensional Ising model and the voter model (on an
arbitrary graph) are prototypical examples of these two situations which are
used here as guiding examples. Defining the fluctuation-dissipation ratio
$X_{ij}(t,t')=\beta R_{ij}/(\partial G_{ij}/\partial t')$, where
$G_{ij}(t,t')=\langle S_i(t)S_j(t')\rangle$ is the spin-spin correlation
function and $\beta$ is a parameter regulating the strength of the perturbation
(corresponding to the inverse temperature when detailed balance holds), we show
that, in the quite general case of a kinetics obeying dynamical scaling, on
equal sites this quantity has a universal form$X_{ii}(t,t') = (t+t')/(2t)$,
whereas $\lim _{t\to \infty}X_{ij}(t,t')=1/2$ for any $ij$ couple. The specific
case of voter models with long-range interactions is thoroughly discussed.
