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We prove a Hurewicz-type theorem for the dynamic asymptotic dimensionoriginally introduced by Guentner, Willett, and Yu. Calculations of (or simplyupper bounds on) this dimension are known to have implications related tocohomology of group actions and the K-theory of their transformation groupC*-algebras. Moreover, these implications are relevant to the currentclassification program for C*-algebras. As a corollary of our main theorem, weshow the dynamic asymptotic dimension of actions by groups on profinitecompletions along sequential filtrations by normal subgroups is subadditiveover extensions of groups, which shows that many such actions by elementaryamenable groups are finite dimensional. We combine this with other novelresults relating the dynamic asymptotic dimension of such an action to theasymptotic dimension of a corresponding box space. This allows us to give upperbounds on the asymptotic dimension of many box spaces (including examples frominfinitely-many groups with exponential growth) using a generalization of theHirsch length for elementary amenable groups. For some of these examples, wecan also find lower bounds by utilizing the theory of ends of groups.

A Hurewicz-type Theorem for the Dynamic Asymptotic Dimension with Applications to Coarse Geometry and Dynamics