  We prove a compactness result for classes of actions of many small categories
on quantum compact metric spaces by Lipschitz linear maps, for the topology of
the covariant Gromov-Hausdorff propinquity. In particular, our result applies
to actions of proper groups by Lipschitz isomorphisms on quantum compact
spaces. Our result provides a first example of a structure which passes to the
limit of quantum metric spaces for the propinquity, as well as a new method to
construct group actions, including from non-locally compact groups seen as
inductive limits of compact groups, on unital C*-algebras. We apply our
techniques to obtain some properties of closure of certain classes of {\gQqcms
s} for the propinquity.
