When you have a station with multiple lines I'm seeing some lines being supplied over others. What are the rules about and how can I manipulate which lines get the goods?

I realize this is from a few months ago, but in case you or anyone else is still looking for an answer, I'll try. I'm not an expert on the game's under-the-hood mechanics, but from what I can tell, it is really based on the strength of downstream demand for the resource.
For instance, I had a lumber mill that I was trying to use to supply both a tool/machine factory in one area and a goods factory in another. The problem was that I had used the tools/machines place for awhile and maxed its production while I did not really do much for the goods factory (wasn't moving them enough to increase demand for new supply), so I really could not get enough planks set aside for the goods factory while piles and piles continued building up for tools/machines that were being produced at high levels and sent everywhere. If I had put in the effort to move more goods, then the factory would have upgraded causing it to demand more planks from the mill which would have helped level out some of the plank distribution. I believe there is a 'potential' value given in the overview/details for each industry which indicates how much of each resource it is currently capable of using (i.e. ordering from the feeder industry) or supplying. Getting the industry to upgrade will increase this and call for greater demand on the resources. Provide more trucks or train cars to move more resources to increase the resources released and cause the industry to upgrade in several months.
The other consideration is that the game is very good at knowing if your line is complete. If there is any interruption, including downstream to the distribution of the end-product, you may see your line shut off from materials. Be sure you have all the trucks, trains, and ships coordinated (with at least one on each line to start the material flow initially).
That's the best I can describe it based on my experiences, anyone else with a better understanding can offer corrections or details. Good luck.