  The off-equilibrium response function $\chi (t,t_w)$ and autocorrelation
function $C(t,t_w)$ of an Ising chain with spin-exchange dynamics are studied
numerically and compared with the same quantities in the case of spin-flip
dynamics. It is found that, even though these quantities are separately
different in the two cases, the parametric plot of $\chi (t,t_w)$ versus
$C(t,t_w)$ is the same. While this result could be expected in higher
dimensionality, where $\chi (C)$ is related to the equilibrium state, it is far
from trivial in the one dimensional case where this relation does not hold. The
origin of the universality of $\chi(C)$ is traced back to the optimization of
domains position with respect to the perturbing external field. This mechanism
is investigated resorting to models with a single domain moving in a random
environment.
