In cellular phenomena, such as cytoplasmic streaming, molecular motors translocate along microtubules carrying cargoes which entrain fluid. The piconewton forces that motors produce can be sufficient to bend or buckle the filaments. When large numbers of such forced filaments interact through the surrounding fluid, as seen in particular stages of oocyte development in Drosophila melanogaster, complex dynamics arise, but the mechanism underlying them has remained unclear. Motivated by these observations, through a combination of theoretical analysis and numerical simulations, we study a simplified microtubules-molecular motor system. Microtubules are modelled as elastic inextensible two-dimensional filaments, and the molecular motor-cargo ensemble as a point force. The analysed dynamics result from the interplay between the forcing, elasticity, and hydrodynamic stresses associated with the motion in a viscous fluid. First, we study a single filament subject to a localised force acting tangentially at its tip. We show that when the external forcing exceeds a finite threshold, the system undergoes a Hopf bifurcation, which results in flapping motion, reminiscent of spermatozoa beating. We elucidate the nature of such instability using a lower-dimensional ‘two-link’ model and linear stability analysis. Then, we generalize the model to describe the real biological system more accurately. In particular, we include the fluid flow created by the molecular motor-cargo ensemble while it is walking along the microtubules, we allow the molecular motor to be located anywhere along the filament, and extend the framework to the case of many motors. Inspired by experiments, in which many filaments interact with one another, we apply these extensions to the multi-filament case, with the aim of studying the collective dynamics of flapping filaments. We consider two distinct scenarios: an array of filaments on a planar wall, and a multitude of filaments inside a sphere. By exploiting asymptotic approximations and parallel computing, we show that an array of filaments can synchronise their motion or reach a final steady bent configuration. Moreover, we shed light on the role of confinement, which proves to be crucial in spontaneously breaking the symmetry in the filament configuration, as experimentally observed. Our results form the basis for the deeper physical understanding of the role of fluid-structure interactions during the oocyte development in Drosophila. By employing a range of models of increasing complexity, we have been able to capture the wave-like filament motion observed in experiments, paving the way for future research involving more physiological details.