Ride-pooling (or ride-sharing) services combine trips of multiple customers along similar routes into a single vehicle. The collective dynamics of the fleet of ride-pooling vehicles fundamentally underlies the efficiency of these services. In simplified models, the common features of these dynamics give rise to scaling laws of the efficiency that are valid across a wide range of street networks and demand settings. However, it is unclear how constraints of the vehicle fleet impact such scaling laws. Here, we map the collective dynamics of capacity-constrained ride-pooling fleets to services with unlimited passenger capacity and identify an effective fleet size of available vehicles as the relevant scaling parameter characterizing the dynamics. Exploiting this mapping, we generalize the scaling laws of ride-pooling efficiency to capacity-constrained fleets. We approximate the scaling function with a queueing theoretical analysis of the dynamics in a minimal model system, thereby enabling mean-field predictions of required fleet sizes in more complex settings. These results may help to transfer insights from existing ride-pooling services to new settings or service locations.