The smaller fragments forming the inner planets will allow them to cool faster than the outer planets, and build a gravitational field more rapidly. As a result, they will have a chance to capture more debris from the supernova cloud than the outer planets will. Due to the close proximity to the sun, there will also be more of the heavier elements present, because the lightest elements get thrown the furthest out during an explosion. Once a blanket of debris surrounds the white dwarf fragment, the cooling process slows—for the layers of rock acts as insulation.
Given a typical 4-inner-planet system, what we find is the innermost planet, Planet 1, will remain mostly “white dwarf,” as being exposed to the heat of the sun will slow the cooling process. Its surface will be composed of the heavy metals (remembering that the white dwarf has an inverse density gradient, and the highest density is on the surface), in a near molten state. Meteoric dust will add a very small quantity to this, as the proximity to the sun will also pull most debris past this small world.
Planets #2 and #3 will cool at a similar rate, and collect a reasonable amount of debris from meteor aggregation. They will be similar in size (based on their fragment size), and collect a reasonable amount of dust and rock on their surfaces. Planet #2 will have a smaller core, but more mantle than Planet #3.
Planet #4, however, being near to the neutral point of the asteroid belt, will pick up some debris, but not nearly as much. It will cool faster than the other three, and will be the first planet of a system capable of harboring life, as the sun will still be in the giant phase, and providing sufficient heat and light for a reasonable, life-bearing environment.
Thus, the size distribution of the inner planets will be: small, medium, medium, small, with planet #4 developing life first, followed by #3, then #2 as the sun moves into the main sequence. Planet #1 will never form the water-based ecosystems that the three other planets will, as the sun will start to get hotter and larger (moving up the main sequence) before the surface of this planet cools sufficiently to retain water in liquid form. This, however, does not preclude the possibility of life based on other ecosystems.
As the sun grows in size and temperature, the inner planets slowly become uninhabitable, succumbing to solar heat, radiation, and charged particles, vaporizing their seas, and creating dry, arid climates.
In our system, Planets #1 through #4 are Mercury, Venus, Earth and Mars. Mars will be the first world to develop water-based life, followed by the Earth, then Venus. By the time Venus moves into the habitable range, Mars will have moved out of it, and Earth will be in its early habitable stage. Each planet’s evolution is unique—Venus has one, short life stage, Earth has one long one, and Mars has two different stages, early and late.