Many-body systems with long-range interactions that exhibit complicated collective dynamics are common in Nature. For example, stellar motion, electrons conducted in materials, charged particles in plasma and sedimentation of particles in a fluid under gravity. Yet, their complex behavior is hard to explain, since each constituent effectively interacts with all the others. A fundamental measure of these dynamics is the fluctuations of physical quantities and their correlations over space or time, where the general challenge is to understand these observables from basic principles. Here we study the dynamics of micro-droplet ensembles flowing in a two-dimensional channel and governed by long-range hydrodynamic dipolar interactions. We measure the fluctuations and spatial correlations of the droplets’ velocities and explain them theoretically from first principles. The velocity fluctuations show a non-monotonous dependence on droplet density. The velocity correlations exhibit a striking spatial structure with both positive and negative peaks that correspond to characteristic modes of collective droplet motion. We show how these velocity fluctuations and correlations are determined by the sum of the basic dipolar interactions over the entire droplet medium. The velocity fluctuations stem from a competition between positive single-droplet contributions and negative contributions of droplet-pairs that arise from excluded-area effects. The velocity spatial correlations are explained by the effect of a single-droplet representing the medium, on a pair of test droplets. The summation over the long-range interactions converges thanks to the low-dimensionality of the system facilitating, for the first time, a theoretical description of the velocity fluctuations and correlations in such a complicated system.