I think you're confused about the structure of the asteroid belt. Pretty much, the asteroids there have been there since the beginning of the solar system. Very few (<1%) of the asteroids in the belt have migrated there in the last few billion years. (There is a suggestion of a small number of asteroids in the very outer belt that moved in during that time)
Asteroids that cross Jupiter's orbit from outside are very rare (in fact they are usually comets) and are on highly eccentric orbits, spending only a small part of their orbit inside Jupiter. When Jupiter gives them the right tweak at the right time, they become what are called Jupiter family comets. There are about 20-100 of them at any time, and the typical lifetime is less than half a billion years; about 6% wind up hitting the sun (or a planet), the rest are ejected from the solar system by interactions with Jupiter.
"On a parallel thought process, once the asteroids pass the Martian orbit they must accelerate towards the Sun."
Why? They continue on in their orbit. The sun isn't a vacuum cleaner, it's just a bit of mass, part of what any object orbits around.