The toroidal grids $C_p\square C_q$ ($C_p\square C_q$ is the Cartesian product of the cyclic graphs of order $p$ and $q$) are graphs $G$ satisfying the local homogeneity property $P(G)$ defined by: given any two points, there exists an isomorphism between the closed neighbourhoods of these points that can be extended to an automorphism of the whole graph $G$. (Why? simply take the translation that maps the first point to the second.)
A generalisation: the local homogeneity property $Q_n(G)$ where given any two points $x$ and $y$ in the graph $G$, there exists an automorphism of G which induces an isomorphism from the subgraph of points of distance $\le n$ from $x$ to the the subgraph of points of distance $\le n$ from $y$. Again, toroidal grids satisfy this property.
My questions:
is there a commonly accepted term for the property $P$ and for the class of graphs with this property?
is there a commonly accepted term for the property $Q_n$ and for the class of graphs with this property?
do any results exist along the lines of classifying finite graphs with these properties?