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In this presentation we study the asymptotic behaviour of infinite systems of coupled linear ordinary differential equations. Each subsystem has identical dynamics that are only dependent on the states of its immediate neigbours. Examples of such systems in particular include the infinite "robot rendezvous problem" and the "platoon system" that are used to approximate the dynamics of large configurations of vehicles. In the presentation introduce novel methods for studying the spectral properties and stability of infinite systems of differential equations. The latter question is particularly interesting due to the fact that the systems in our class are known to lack uniform exponential stability. As our main results, we introduce general conditions for strong stability and derive rational rates of convergence for the solutions using recent results in the theory of nonuniform stability of strongly continuous semigroups.