Abstract We have studied the capture of planetesimals in Trojan orbits by a growing proto-Jupiter by integrating numerically the equations of motion of planetesimals in a four-body problem Sun–Jupiter–Saturn–planetesimal. The masses of Jupiter and Saturn increase exponentially with time. Nebular gas drag on small planetesimals is taken into account. We show that a fraction of the planetesimals near Jupiter's orbit and almost all planetesimals originally in horseshoe orbits are trapped into stable Trojan orbits. This mechanism was probably an important factor together with collisional diffusion, for delivering a large amount of planetesimals into Trojan orbits during the final growth of Jupiter by gas in-fall. We find also that the libration amplitude of trapped Trojans is damped by the increase of Jupiter's mass reinforcing the stability of the resonance lock. For planetesimals small enough to be perturbed by nebular gas drag, we find an asymmetry between the trapping rate in L5 compared to L4. Peale (1993, Icarus 106, 308–322) noted also that the stability of the L4 resonant region is reduced in presence of an intense frictional drag. We present here a simple semi-analytical explanation for the trapping asymmetry based on the evolution of the angular difference between the planetesimal's and proto-Jupiter's longitude of perihelion. The asymmetry between L4 and L5 could have led to the capture of a larger number of small planetesimals in L5. This may have favored a higher degree of collisional activity in L5 and might explain an alleged asymmetry in the size distribution of the two swarms.