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The civilian shipping lines are able to move installations for you. You can setup a supply or demand contract for a specified amount of a specified installation type using a new section on the Civilian tab of the Economics window. For example, you may setup a supply contract for 20 mines on Earth and a demand contract for 20 mines on Mars. For the civilians, this works in a very similar way to trade goods. They will look for matching supply and demand contracts, in this case on Earth and Mars, and create a trade run. As soon as they select the trade run and setup their orders, one installation is deducted from both the supply and demand contracts. This is to prevent duplication. The new contract section will show outstanding contracts and any contracts in progress that affect the population you are viewing. You don't have to worry about matching up contracts as the civilians will figure that out by themselves. If you have a surplus of something on a planet, then create a supply contract. If you need something, setup a demand contract. There may not be a match for it immediately but as soon as one appears the civs will spot it. Each freighter run will cost you 5 wealth for a same system contract and 10 wealth for an interstellar contract, which is paid when the freighter completes the contract. This wealth will be added to the wealth of the shipping line.
When a freighter is looking for a new trade run it will look at government contracts at its current location first, then potential trade runs at its current location, then government contracts originating in the same system, then trade runs originating in the same system, etc. As you already know, civilians work out trade runs for themselves and trade routes and trade hubs develop. In a well-developed game, if the civilian trade routes are not going near the locations of the supply contracts because there are easier trade runs available, you may have to move your installations to the nearest trade hub. It will be up to you to decide if partial civilian shipping is better than making the whole run with your own ships.
Unlike trade goods, all government contracts are within one Empire only. Civilians from one empire will not look for matching demands between their Empire and another one.