  In this paper, we study the asymptotic behavior of solution to a
non-autonomous diffusion equations with delay containing some hereditary
characteristics and nonlocal diffusion in time-dependent space
$C_{\mathcal{H}_{t}(\Omega)}$. When the nonlinear function $f$ satisfies the
polynomial growth of arbitrary order $p-1$ $(p \ge 2)$ and the external force
$h \in L_{l o c}^{2}\left(\mathbb{R} ; H^{-1}(\Omega)\right)$, we establish the
existence and regularity of the time-dependent pullback attractors.
