Hydrodynamical simulations predict the inward migration of giant planets during the gas phase of the protoplanetary disc. This phenomenon is also invoked to explain resonant and near-resonant exoplanetary system structures. The early inward migration may also have affected our Solar System and sculpted its different minor planet reservoirs. In this study we explore how the early inward migration of the giant planets shapes the Kuiper belt. We test different scenarios with only Neptune and Uranus and with all the four giant planets, also including some models with the subsequent outward planetesimal-driven migration of Neptune after the gas dispersal. We find objects populating mean motion resonances even when Neptune and Uranus do not migrate at all or only migrate inwards. When the planets are fixed, planetesimals stick only temporarily to the mean motion resonances, while inwards migration yields a new channel to populate the resonances without invoking convergent migration. However, in these cases, it is hard to populate mean motion resonances that do not cross the planetesimal disc (such as 2:1 and 5:2) and there is a lack of resonant Kuiper belt objects (KBOs) crossing Neptune's orbit. These Neptune crossers are an unambiguous signature of the outward migration of Neptune. The starting position and the growth rate of Neptune have consequences for the contamination of the classical Kuiper belt region from neighbouring regions. The eccentricity and inclination space of the hot classical Kuiper belt objects and the scattered disc region become much more populated when all the giant planets are included. The 5:2 resonance with Neptune becomes increasingly populated with deeper inward migrations of Neptune. However, the overall inclination distribution is still narrower than suggested by observations, as is generally the case for Kuiper belt population models.