Due to the sub-millimetre dimensions and accessibility of the distal regions of the lung, respiratory flows in the pulmonary acinus are often difficult to assess. However, a realistic description of acinar flows is needed to understand aerosol transport and deposition for medical applications such as aerosol inhalation. In an effort to develop more realistic computational fluid dynamics (CFD) models of the pulmonary acinus, we have simulated convective flows under rhythmic breathing motion in a space-filling model of an acinar branching tree. Our model captures well the variety of 3D flow patterns present along the tree and confirms the existence of complex recirculating alveolar flows. Our results emphasize the role of the alveolar to ductal flow ratio in characterizing acinar flows. Lagrangian particle trajectories suggest that massless particles, not influenced by sedimentation or diffusion, stay principally confined within acinar ducts without entering into alveoli. The inherent modularity of the present model is well suited to create more complete geometries of acinar trees and investigate the influence of convective acinar flows on realistic sedimenting and/or diffusing particles.