  We extend the Gromov-Hausdorff propinquity to a metric on Lipschitz dynamical
systems, which are given by strongly continuous actions of proper monoids on
quantum compact metric spaces via Lipschitz morphisms. We prove that our
resulting metric is zero between two Lipschitz dynamical systems if and only if
there exists an equivariant full quantum isometry between. We also present
sufficient conditions for Cauchy sequences to converge for our new metric, thus
exhibiting certain complete classes of Lipschitz dynamical systems. We apply
our work to convergence of the dual actions on fuzzy tori to the dual actions
on quantum tori. Our framework is general enough to also allow for the study of
the convergence of continuous semigroups of positive linear maps and other
actions of proper monoids.
