Here's the short version:

• Use the default settings. Don't change them.
  
• Connect two districts with a District Crossing.
  
• Build storage for the items you want in the districts you want those items in.

That's it. That's all you really need in 99% of cases.

Goods after Update 4 flow like water.

If you have a district that has many items in storage, and a connected district that has a lot of empty storage of that same item, beavers will attempt to move items from the filled district to the empty district until both districts have the same percentage of storage filled in both districts.

So, for example:

District One might have storage for 200 Gears.
District Two might have storage for 40 Gears.

If District One's storage is full, so they have 200 Gears, and District Two has no gears, beavers will move Gears around until District One is at around 167 Gears and District Two is around 33 Gears, so that both districts are at around 87% full in their available Gears storage. One district was at 100%, the other was at 0%, now they're both at (around) 87%.

Note: Only storage buildings count as available storage. For example, trying to build a smelter when you don't have scrap metal storage in the same district (or don't have that district set to Import Always on scrap) won't suddenly pull scrap into the district. A building looking to make planks out of logs isn't going to create import demand for logs: only a building that stores logs will do that.

The District Storage building can also act as a storage building for the behavior I described above, but only for goods set to "Import always", which is how the default settings will get your initial products moved into a district so you can build storage for other goods. A setting of "Import if needed" will still allow goods to enter that District Crossing to be moved into the district; it just won't count that storage for the purposes of deciding whether items need to enter the district in the first place.

The District Crossing itself is a storage building, and one that always has room for 30 of every item. That space is shared between the two sides of the building.

However, this storage only "counts" for the purpose of balancing goods between districts IF you have that item set to Import Always.

Setting an item to "Import Always" is like telling the game "please consider every District Crossing in this district like it's an additional 30 units of space for that item".

Items will balance accordingly.

This is why the default settings are ideal for most situations. By default, water, logs, planks, and every item that addresses the Hunger stat (so every edible food, but not coffee) are set to "Import always". Everything else is set to "Import if needed".

So a brand new district built with one District Crossing attached with default settings will behave like it has 30 storage for:

• Logs
  
• Planks
  
• Water
  
• All hunger-curing food

Which is everything you might possibly need to keep a already healthy beaver alive, and build more storage to bring more goods in.

If an item is set to "Import if needed" rather than "Import always", then only storage buildings other than District Crossings count as storage space. The items can still be moved into a District Crossing, but they'll only do so if you have other storage waiting for those goods to be moved to.

To put that another way, if you have zero storage for metal blocks, and metal blocks are set to "Import if needed", then the game still acts as if you have zero storage for metal blocks. It won't move metal blocks into that district.

If you set that District to "Import Always" on metal blocks, each District Crossing in that district now has 30 storage for those metal blocks. Adjacent districts will attempt to balance this new storage space with the metal blocks they already have in their connected districts.

If you leave that district's metal blocks setting to "Import if needed", and then build storage for 20 metal blocks, that district is considered to have space for 20 metal blocks, and will balance with adjacent districts accordingly. The metal blocks will move 'through' the 30 spaces in any relevant District Crossings to get them into your 20 storage, but the 30 spaces in the District Crossing are not part of the balancing equation.

Lets say you have three districts: Mine, Middle, and Refine.

District Mine mines Scrap Metal.
District Refine refines Scrap Metal into Metal Blocks.

District Middle connects the two districts. Mine<->Middle<->Refine.

You have storage for scrap at Refine, but no storage in Middle. Mine still isn't sending goods. Why?

Mine can't send Scrap directly to Refine. It has to go through Middle. If Middle doesn't need Scrap, Scrap won't be sent to Middle, so it can never reach Refine.

How do you fix this?

You have two options: Build storage in Middle so that Mine knows to send Scrap to Middle (so it can get to Refine) or...

... set the Distribution Settings for District Middle to Import Always on Scrap.

Now Scrap will always be sent from Mine to Middle, even if Middle doesn't have storage available. Then, once the Scrap is in Middle, Refine can receive it (because they have storage available).