Dust happens in two ways. "Primordial dust" just condenses out of the protostellar material in the disc providing it gets cool enough and dense enough. "Second generation" dust is generated by collisions between planetesimals.
In both cases there are a number of mechanisms that get rid of this dust, two of the main ones being radiation pressure and Poynting-Robertson drag.
Radiation pressure (the force with which radiation from the protosun acts upon a dust particle) will vary with the flux of radiation, and falls as $r^{-2}$. This has the same dependence as gravitational force on the dust and so the ratio of the two forces will be approximately constant. The ratio will depend on the density, albedo and shape of the dust particles. However, generally speaking, anything smaller than $10^{-7}$ m will get blown out of the Solar System on a very short (ballistic) timescale.
Thus, what happens is that as soon as the Solar System becomes optically thin to the Sun's radiation then all the small primordial dust particles are blown away. Of course this dust can be regenerated by collisions between planetesimals, but again, this will be an equilibrium process in the sense that small dust particles would need to be being continuously generated in order for a population to be present, since they get blown out of the Solar System quickly.
Poynting Robertson drag is a tangential acceleration of the dust particles caused by the fact that in their frame of reference, the radiation from the Sun does not arrive radially. It slows the dust particles down and they spiral in to the Sun. Again, this will act as soon as dust particles are being illuinated by the protosun.
Poynting Robertson drag also depends on the density of the particles, their size and the distance to the Sun as $r^{-5/2}$. As a result, it gets more important for dust at smaller $r$. Poynting-Robertson drag is irrelevant for small particles, since they are blown away by radiation pressure. For larger particles, they will spiral into the Sun on a timescale that depends on their size and starting position. For dust of size $\sim 10^{-6}$ m at $r=1$ au, that timescale is only 10,000 years. It is longer for dust that is further out and for larger particles. The net result is that it is easy to clear out the inner disc of dust, less easy to get rid of it further out.
Dust in the outer Solar System that survives radiation pressure ($>10^{-7}$ m) and Poynting Robertson drag ($>$ a few au) may survive for a long time, but it will be subject to being captured by growing giant planets. It will also eventually be pushed out of the Solar System by the influence of the solar wind - an outflow that is distinct from radiation pressure. Stellar winds/outflows in young stars are probably much stronger than the weak solar wind we have today and small dust particles are well-coupled to the outflowing gas. Some of the larger particles 10-100 microns might survive for tens of millions of years, but ultimately almost all the dust seen in the Solar System after that is second generation - formed in collisions.
A further source of dust destruction is radial infall for small dust particles that are well-coupled to gas but which is being accreted onto the star during its first few million years. Larger dust particles may be less well-coupled but these can drift inwards because of friction between the dust and the gas that it moves through. This can then leave it susceptible to Poynting-obertson drag as before.
Empirically, we observe that the process of dust removal appears to be complete in 10-20 million years and any dust that is seen thereafter is episodic and probably the result of collisions between planitesimals.
