The evolution of molecular clouds in galactic centres is thought to differ from that in galactic discs due to a significant influence of the external gravitational potential. We present a set of numerical simulations of molecular clouds orbiting on the 100-pc stream of the Central Molecular Zone (the central of the Galaxy) and characterize their morphological and kinematic evolution in response to the background potential and eccentric orbital motion. We find that the clouds are shaped by strong shear and torques, by tidal and geometric deformation, and by their passage through the orbital pericentre. Within our simulations, these mechanisms control cloud sizes, aspect ratios, position angles, filamentary structure, column densities, velocity dispersions, line-of-sight velocity gradients, spin angular momenta, and kinematic complexity. By comparing these predictions to observations of clouds on the Galactic Centre ‘dust ridge’, we find that our simulations naturally reproduce a broad range of key observed morphological and kinematic features, which can be explained in terms of well-understood physical mechanisms. We argue that the accretion of gas clouds on to the central regions of galaxies, where the rotation curve turns over and the tidal field is fully compressive, is accompanied by transformative dynamical changes to the clouds, leading to collapse and star formation. This can generate an evolutionary progression of cloud collapse with a common starting point, which either marks the time of accretion on to the tidally compressive region or of the most recent pericentre passage. Together, these processes may naturally produce the synchronized starbursts observed in numerous (extra)galactic nuclei.