  In this paper, we consider the asymptotic behavior of weak solutions for
non-autonomous diffusion equations with delay in time-dependent spaces when the
nonlinear function $f$ is critical growth, the delay term $g(t, u_t)$ contains
some hereditary characteristics and the external force $h \in L_{l o
c}^{2}\left(\mathbb{R} ; L^{2}(\Omega)\right)$. Firstly, we prove the
well-posedness of solutions by using the Faedo-Galerkin approximation method.
Then after a series of elaborate energy estimates and calculations, we
establish the existence and regularity of pullback attractors in time-dependent
spaces $C_{\mathcal{H}_{t}(\Omega)}$ and $C_{\mathcal{H}^{1}_{t}(\Omega)}$
respectively.
