  In this paper, under some appropriate assumptions, we prove the existence of
the minimal time-dependent pullback $\mathcal
D_{\sigma}^{\mathcal{H}_{t}}$-attractors ${\mathcal{A}}_{\mathcal
D_{\sigma}^{\mathcal{H}_{t}}}$ for the non-autonomous nonlocal diffusion
equations in time-dependent space $\mathcal{H}_{t}(\Omega)$. Next, in same
phase space, using a priori estimate and energy methods we establish the
existence of time-dependent pullback attractors $\left\{A_{\xi}(t)\right\}_{t
\in \mathbb R}$ and the upper semicontinuity of $\left\{A_{\xi}(t)\right\}_{t
\in \mathbb R}$ and the global attractor $A$ of the equation with $\xi = 0$,
that is, $$ \lim_{\xi \rightarrow 0^{+}} \operatorname{dist}_{\mathcal
H_{t}}\left(A_{\xi}(t), A \right)=0. $$
