Let $X$ be a metric space and let $J$ be a subgroup of $\text{Isom}(X)$. For any $x \in X$ and compact subset $K \subset X$, consider the set $$A = \left\{ g \in J : g(x) \in K \right\}.$$ What conditions on $X$ and/or J are necessary/sufficient in order to conclude that $A \subset \text{Isom}(X)$ is compact? (Here $\text{Isom}(X)$ carries the compact-open topology).
For example, I know $A$ is compact when $X = \mathbb{H}^2$ and $J = \text{SL}(2,\mathbb{R})$ or $J = \text{PSL}(2,\mathbb{R})$.
Given your examples, I assume that you are mostly interested in proper metric spaces, i.e. metric spaces where a subset if compact if and only if it is closed and bounded. The most natural topology on the group of isometries $Isom(X)$ in this case is the topology of uniform convergence on compact subsets. (Equivalently, you can equip $Isom(X)$ with topology of pointwise convergence.)
You probably also want to consider closed subgroups $J< Isom(X)$.
It is an application of Arzela-Ascoli theorem (combined with the diagonal argument) that in this setting the subset $$ A_{J,K,x} = \left\{ g \in J : g(x) \in K \right\}$$ is compact in $Isom(X)$ and, hence (since $J$ is closed), in $J$.