Jupiter is the heaviest body in the solar system after the sun, so it mostly affects other bodies rather than being affected much itself. To lower it's orbit and cross the asteroid belt, it would have to lose orbital energy. The most plausible way is by transferring this energy to saturn, which in turn would have to go on a higher orbit. Due to the near 2:5 resonance of Jupiter and Saturn, they regulary interact with each other and exchange small amounts of orbital energy. If Saturn is in front of Jupiter at a close encounter, Jupiter gains energy, if Saturn is behind, Jupiter loses energy to Saturn. These energy transfers almost cancel each other out and would completely cancel each other out, if they were on a exact 2:5 resonance. However, they are not, and the chaotic nature of the general 3 body problem makes exact predictions for very long term stability difficult.
Another way of gaining or losing orbital energy is rotation and tidal forces of the center object. For example, the earth slowly loses rotational energy by tidal forces to the moon, therefore the moon's orbit moves away from us. However, regarding the suns rotation energy transferring to Jupiter, this effect is negligible compared to the interactions with saturn.