  In this paper we continue our earlier investigations into the asymptotic
behaviour of infinite systems of coupled differential equations. Under the mild
assumption that the so-called characteristic function of our system is
completely monotonic we obtain a drastically simplified condition which ensures
boundedness of the associates semigroup. If the characteristic function
satisfies certain additional conditions we deduce sharp rates of convergence to
equilibrium. We moreover address the important and delicate issue of the role
of the infinite system in understanding the asymptotic behaviour of large but
finite systems, and we provide a precise way of obtaining size-independent
rates of convergence for families of finite-dimensional systems. Finally, we
illustrate our abstract results in the setting of the well-known platoon
problem.
