  The measurement of the $21$ cm signal from the Cosmic Dawn is a major goal
for several existing and upcoming radio interferometers such as NenuFAR and the
SKA. During this era before the beginning of the Epoch of Reionization, the
signal is more difficult to observe due to brighter foregrounds but reveals
additional information on the underlying astrophysical processes encoded in the
spatial fluctuations of the spin temperature of hydrogen. To interpret future
measurements, controlling the level of accuracy of the Lyman-$\alpha$ flux
modelling is mandatory. In this work, we evaluate the impact of various
approximations that exist in the main fast modelling approach compared to the
results of a costly full radiative transfer simulation. The fast SPINTER code,
presented in this work, computes the Lyman-$\alpha$ flux including the effect
of wing scatterings for an inhomogeneous emissivity field, but assuming an
otherwise homogeneous expanding universe. The LICORICE code computes the full
radiative transfer in the Lyman-$\alpha$ line without any substantial
approximation. We find that the difference between homogeneous and
inhomogeneous gas density and temperature is very small for the computed flux.
On the contrary, neglecting the effect of gas velocities produces a significant
change in the computed flux. We identify the causes (mainly Doppler shifts due
to velocity gradients) and quantify the magnitude of the effect in both an
idealised setup and a realistic cosmological situation. We find that the
amplitude of the effect, up to a factor of $\sim 2$ on the $21$ cm signal power
spectrum on some scales (depending on both other model parameters and the
redshift), can be easily discriminated with an SKA-like survey and already be
approached, particularly for exotic signals, by the ongoing NenuFAR Cosmic Dawn
Key Science Program.
