  The dynamical degrees of a rational map $f:X\dashrightarrow X$ are
fundamental invariants describing the rate of growth of the action of iterates
of $f$ on the cohomology of $X$. When $f$ has nonempty indeterminacy set, these
quantities can be very difficult to determine. We study rational maps
$f:X^N\dashrightarrow X^N$, where $X^N$ is isomorphic to the Deligne-Mumford
compactification $\overline {\mathcal M}_{0,N+3}$. We exploit the stratified
structure of $X^N$ to provide new examples of rational maps, in arbitrary
dimension, for which the action on cohomology behaves functorially under
iteration. From this, all dynamical degrees can be readily computed (given
enough book-keeping and computing time). In this article, we explicitly compute
all of the dynamical degrees for all such maps $f:X^N\dashrightarrow X^N$,
where $\mathrm{dim}(X^N)\leq 3$ and the first dynamical degrees for the
mappings where $\mathrm{dim}(X^N)\leq 5$. These examples naturally arise in the
setting of Thurston's topological characterization of rational maps.
