The initial falling is pretty well covered by the other answers, so this will focus on the other question in the OP.
As the trash collects over thousands of years, would it start to pile up more in some places then others?
Many of the processes would apply to trash on Junkyard the same way that they apply to dirt on Earth.
Precipitation
If you wanted there to be a variety in its distribution, you could add weather that would prevent some trash from making it to the surface, or dissolve trash, creating a climate-like distribution of trash. For example, if your planet had acidic oceans, the patterns of accelerated trash decay would form in the same way that patterns of precipitation form on Earth, with less precipitation corresponding to less disintegration, and thus more trash. 208
The oceans themselves would only contain trash that could survive their acidity.
For example, if Junkyard had Earth's geography, trash on the Western side of the Rockies would receive a lot more acid rain than trash on the opposite side, analogous to how California receives a lot more rain than Nevada.
Geological Activity
If the Junkyard planet is geologically active, subduction zones might swallow up a lot of trash, and the areas where new crust is made would be briefly bare. Also, the way trash reacts to lava could cause areas near volcanos to have unusual concentrations of specific types of trash
Weathering from Wind
Additionally, if certain latitudes have faster wind than others (comparable to the winds around Antarctica here on Earth), that trash would erode faster (on a very slow time scale, but still), and would have a tendency to settle around the belt instead of inside of it.
Glaciers
I'm sure there are more creative ways to incorporate glacial erosion, but at the very least, if someone dropped a comet onto the planet, its chunks might move like glaciers, carving out valleys behind them.
For processes that wouldn't apply to Earth:
Tidal Locking
If the planet orbits around a star, you might want to make it tidally locked, causing trash on one side to melt, and causing trash on the other side to accumulate. Additionally, dumpers may prefer one side or the other to avoid the heat of the star, or to get its energy, depending on which side you prefer. 208