In the eternal domination game played on graphs, an attacker attacks a vertex at each turn and a team of guards must move a guard to the attacked vertex to defend it. The guards may only move to adjacent vertices on their turn. The goal is to determine the eternal domination number $\gamma^{\infty}_{all}$ of a graph, which is the minimum number of guards required to defend against an infinite sequence of attacks. This paper first continues the study of the eternal domination game on strong grids $P_n\boxtimes P_m$. Cartesian grids $P_n \square P_m$ have been vastly studied with tight bounds existing for small grids such as $k\times n$ grids for $k\in \{2,3,4,5\}$. It was recently proven that $\gamma^{\infty}_{all}(P_n \square P_m)=\gamma(P_n \square P_m)+O(n+m)$ where $\gamma(P_n \square P_m)$ is the domination number of $P_n \square P_m$ which lower bounds the eternal domination number [Lamprou et al. Eternally dominating large grids. Theoretical Computer Science, 794:27-46, 2019]. We prove that, for all $n,m\in \mathbb{N^*}$ such that $m\geq n$, $\lfloor \frac{n}{3} \rfloor \lfloor \frac{m}{3} \rfloor+\Omega(n+m)=\gamma_{all}^{\infty} (P_{n}\boxtimes P_{m})=\lceil \frac{n}{3} \rceil \lceil \frac{m}{3} \rceil + O(m\sqrt{n})$ (note that $\lceil \frac{n}{3} \rceil \lceil \frac{m}{3} \rceil$ is the domination number of $P_n\boxtimes P_m$). We then generalise our technique to prove that $\gamma_{all}^{\infty}(G)=\gamma(G)+o(\gamma(G))$ for all graphs $G\in \mathcal{F}$, where $\mathcal{F}$ is a large family of $D$-dimensional grids which are supergraphs of the $D$-dimensional Cartesian grid and subgraphs of the $D$-dimensional strong grid. In particular, $\mathcal{F}$ includes both the $D$-dimensional Cartesian grid and the $D$-dimensional strong grid.