  We establish two main results for the asymptotic dimension of countable
approximate groups. The first one is a Hurewicz type formula for a global
morphism of countable approximate groups $f:(\Xi, \Xi^\infty) \to (\Lambda,
\Lambda^\infty)$, stating that $\mathrm{asdim} \Xi \leq \mathrm{asdim} \Lambda
+\mathrm{asdim} ([\mathrm{ker} f]_c)$. This is analogous to the
Dranishnikov-Smith result for groups, and is relying on another Hurewicz type
formula we prove, using a 6-local morphism instead of a global one. The second
result is similar to the Dranishnikov-Smith theorem stating that, for a
countable group $G$, $\mathrm{asdim} G$ is equal to the supremum of asymptotic
dimensions of finitely generated subgroups of $G$. Our version states that, if
$(\Lambda, \Lambda^\infty)$ is a countable approximate group, then
$\mathrm{asdim} \Lambda$ is equal to the supremum of asymptotic dimensions of
approximate subgroups of finitely generated subgroups of $\Lambda^\infty$, with
these approximate subgroups contained in $\Lambda^2$.
