It may be a bit more like the Triangle Trades of old [0].
Raw/rare materials and elements are mined in the Belt and are then sent to Earth for use in manufacture. You pick up the high quality goods and tools from Earth and ship them to Mars/Moon. Since there is water on Mars/Moon, you grow food, 'wet' supplies, and unfinished goods in the weaker gravity well for shipment to the Belt. These arrive in the Belt and you repeat the process, profiting at every spaceport.
As the metropole has overwhelming manpower/firepower due to it's high class manufacturing base and population, the threat of violence is just plain bad for business all around and is discouraged.
Now I'm gong to speculate very hard: Eventually, the lesser manufacturing base (Moon/Mars) will gain in economic and political power as their frontier life becomes more civilized and independent of the metropole. And they may try to declare independence fully. However, in the case of Moon/Mars, you're looking at a fully independent multi-generational extraterrestrial biospheres at that point and probably millions of people, if not 10s of millions, to run the whole thing. Lots of people that have never been to Earth and never plan on going. I really don't see that much trade/mining would be needed to go to Earth to build up and support permanently independent biospheres of a few generations of native Martians/Moonies.
[0] https://en.wikipedia.org/wiki/Triangular_trade