Aims. Assuming fast radio bursts (FRBs) are produced by matter travelling ultra-relativistically in a localised region of a smooth bundle of streamlines, we study the constraints applied by geometry to the morphology and polarisation of the burst in time and frequency independently of the intrinsic radiative process. Methods. We express the problem only in terms of the local properties of direction and curvature of a streamline. This allows us to cast the general results to any desired geometry. We illustrate by applying this framework to two geometries inspired by pulsar and magnetar magnetospheres, namely the dipolar polar-cap region and a magnetic dipole with an additional toroidal component. Results. Geometry constrains bursts to occur within an envelope in the frequency vs. time plane (dynamic spectrum). This envelope notably characterises spectral occupancy and frequency drifts (both burst-to-burst and within an individual burst). We illustrate how one can simulate bursts by specifying some basic properties of an intrinsic emission process. In particular we show that the typical properties of one-off bursts can be produced in polar-cap geometry by a star with spin period > 1s, while bursts from repeating sources are better accounted for with an additional strong toroidal component and a sub-second spin period. Conclusions. We propose that a relationship between burst morphologies and the properties of the source, such as its spin period and magnetospheric properties, can be established at least qualitatively based on geometrical considerations. Our results favour models where repeaters are younger and faster magnetars with highly twisted magnetospheres.