Many asteroid families are identified and well characterized all over the main asteroid belt. Interestingly, however, none of them are older than 4 Gyr. Many mechanisms have been proposed to disperse such old primordial asteroid families that presumably have existed, but only very few have really worked. Here we present a plausible mechanism for dispersing primordial asteroid families that is based on the 5-planet instability model known as jumping Jupiter. Using two different evolutions for the jumping-Jupiter model, we have numerically integrated orbits of eight putative primordial families. Our results show that the most important effect on the asteroid families' eccentricity and inclination dispersal is that of the secular resonances, in some cases associated with the mean motion resonances. As for the semimajor axes spreading we find that the principal effect is that of close encounters with the fifth giant planet whose orbit briefly overlaps with (part of) the main belt. Therefore, the existence of a fifth giant planet with the mass comparable with that of Uranus' or Neptune's could contribute in important ways to dispersal of the primordial asteroid families. To have that effect, the interloper planet should go into and considerably interact with the asteroids during the instability phase.