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We use $Λ$CDM cosmological hydrodynamical simulations to explore the kinematics of gaseous discs in late-type dwarf galaxies. We create high-resolution 21-cm 'observations' of simulated dwarfs produced in two variations of the EAGLE galaxy formation model: one where supernova-driven gas flows redistribute dark matter and form constant-density central 'cores', and another where the central 'cusps' survive intact. We 'observe' each galaxy along multiple sight lines and derive a rotation curve for each observation using a conventional tilted-ring approach to model the gas kinematics. We find that the modelling process introduces systematic discrepancies between the recovered rotation curve and the actual circular velocity curve driven primarily by (i) non-circular gas orbits within the discs; (ii) the finite thickness of gaseous discs, which leads to overlap of different radii in projection; and (iii) departures from dynamical equilibrium. Dwarfs with dark matter cusps often appear to have a core, whilst the inverse error is less common. These effects naturally reproduce an observed trend which other models struggle to explain: late-type dwarfs with more steeply-rising rotation curves appear to be dark matter-dominated in the inner regions, whereas the opposite seems to hold in galaxies with core-like rotation curves. We conclude that if similar effects affect the rotation curves of observed dwarfs, a late-type dwarf population in which all galaxies have sizeable dark matter cores is most likely incompatible with current measurements.