We consider models of $N$ interacting objects, where the interaction is via a common resource and the distribution of states of all objects. We consider the case where the number of transitions per time slot per object vanishes as $N$ grows. We show that, under mild assumptions and for large $N$, the occupancy measure converges, in probability and in mean square over any finite horizon, to a deterministic dynamical system. Our method of proof is inspired by stochastic approximation algorithms. The convergence results allow us to derive properties valid in the stationary regime. We use this to develop a critique of the fixed point method sometimes used in conjunction with the decoupling assumption.