  We prove a Hurewicz-type theorem for the dynamic asymptotic dimension
originally introduced by Guentner, Willett, and Yu. Calculations of (or simply
upper bounds on) this dimension are known to have implications related to
cohomology of group actions and the K-theory of their transformation group
C*-algebras. Moreover, these implications are relevant to the current
classification program for C*-algebras. As a corollary of our main theorem, we
show the dynamic asymptotic dimension of actions by groups on profinite
completions along sequential filtrations by normal subgroups is subadditive
over extensions of groups, which shows that many such actions by elementary
amenable groups are finite dimensional. We combine this with other novel
results relating the dynamic asymptotic dimension of such an action to the
asymptotic dimension of a corresponding box space. This allows us to give upper
bounds on the asymptotic dimension of many box spaces (including examples from
infinitely-many groups with exponential growth) using a generalization of the
Hirsch length for elementary amenable groups. For some of these examples, we
can also find lower bounds by utilizing the theory of ends of groups.
