Generally, setting contracts is the way to grow civ shipping most quickly. My observation has been that civs will prioritize installation contracts over the shorter distances first as these are most profitable, so the best way to handle contracts is to set them up as short-term contracts until they are filled. If you demand 5,000 infrastructure for Luna and 5,000 for Mars, the civs will tend to ship 5,000 units to Luna before shipping any to Mars - which might be okay if you have time to wait (and will actually grow the civs quicker due to higher profits), but if you want to get stuff to both colonies more evenly it will be more effective to demand maybe 200 infra at a time in each colony and wait for those orders to be completely filled.

For shipments to outlying systems I don't have enough good observations of the civ shipping logic but it should be more even since the profit margins are based on the number of systems between endpoints. Additionally, your civ freighters should become dispersed over time which will also help to make the contract selection more balanced and/or "random" in appearance.

Contracts always have higher priority than trade good shipping, which may be another argument in favor of issuing more frequent smaller contracts as your civs can ship trade goods in between contracts.