In coverage control problems that involve time-varying density functions, the coverage control law depends on spatial integrals of the time evolution of the density function. The latter is often neglected, replaced with an upper bound or calculated as a numerical approximation of the spatial integrals involved. In this paper, we consider a special case of time-varying density functions modeled as Gaussian Mixture Models (GMMs) that evolve with time via a set of time-varying sources (with known corresponding velocities). By imposing this structure, we obtain an efficient time-varying coverage controller that fully incorporates the time evolution of the density function. We show that the induced trajectories under our control law minimise the overall coverage cost. We elicit the structure of the proposed controller and compare it with a classical time-varying coverage controller, against which we benchmark the coverage performance in simulation. Furthermore, we highlight that the computationally efficient and distributed nature of the proposed control law makes it ideal for multi-vehicle robotic applications involving time-varying coverage control problems. We employ our method in plume monitoring using a swarm of drones. In an experimental field trial we show that drones guided by the proposed controller are able to track a simulated time-varying chemical plume in a distributed manner.