Jupiter Trojans are a resonant asteroidal population characterised by photometric colours that are compatible with trans-Neptunian objects, high inclinations, and an asymmetric distribution of the number of asteroids between the two swarms. Different models have been proposed to explain the high inclination of the Trojans and to interpret their relation with the Trans-Neptunian objects, but none of these models can also satisfactorily explain the asymmetry ratio between the number of asteroids in the two swarms. It has recently been found that the asymmetry ratio can arise if Jupiter has migrated inwards through the protoplanetary disc by at least a few astronomical units during its growth. The more numerous population of the leading swarm and the dark photometric colours of the Trojans are natural outcomes of this new model, but simulations with massless unperturbed disc particles led to a flat distribution of the Trojan inclinations and a final total mass of the Trojans that was 3-4 orders of magnitude larger than the current mass. We here investigate the possible origin of the peculiar inclination distribution of the Trojans in the scenario where Jupiter migrates inwards. We analyse different possibilities: (a) the secular evolution of an initially flat Trojan population, (b) the presence of planetary embryos among the Trojans, and (c) capture of the Trojans from a pre-stirred planetesimal population in which Jupiter grows and migrates. We find that the secular evolution of the Trojans and secular perturbations from Saturn do not affect the inclination distribution of the Trojans appreciably, nor is there any significant mass depletion over the age of the Solar System. Embryos embedded in the Trojan swarms, in contrast, can stir the Trojans to their current degree of excitation and can also deplete the swarms efficiently, but it is very difficult to remove all of the massive bodies in 4.5 Gyr of evolution. We propose that the disc where Jupiter's core was forming was already stirred to high inclination values by other planetary embryos competing in the feeding zone of Jupiter's core. We show that the trapped Trojans preserve their high inclination through the gas phase of the protoplanetary disc and that Saturn's perturbations are more effective on highly inclined Trojans, leading to a lower capture efficiency and to a substantial depletion of the swarms during 4.5 Gyr of evolution.