This paper investigates the asymptotic behavior of solutions to certain in finite systems of ordinary differential equations. In particular, we use results from ergodic theory and the asymptotic theory of C-0-semigroups to obtain a characterization, in terms of convergence of certain Cesaro averages, of those initial values which lead to convergent solutions. Moreover, we obtain estimates on the rate of convergence for solutions whose initial values satisfy a stronger ergodic condition. These results rely on a detailed spectral analysis of the operator describing the system, which is made possible by certain structural assumptions on the operator. The resulting class of systems is sufficiently broad to cover a number of important applications including, in particular, both the so-called robot rendezvous problem and an important class of platoon systems arising in control theory. Our method leads to new results in both cases.