So the pallet warehouse works like a productions facility. It will accept an input from either distribution or the manual loading zone. Being a productions facility you can set it to store, distribute or sell. To answer you question, if you wanted to move say your flour from your warehouse to your bakery; you simply set flour to distribute, inside the warehouse productions tab, each cycle it will send flour to your bakery the same way your grain mill would if you set it to distribute. If you actually wanted to manually grab say 4 pallets from the warehouse this is more difficult, items that appear in the manual pickup zone appear in the same order they have been loaded in, so if you put in 100 pallets of eggs and then 2 pallets of flour, you would have to manually grab all 100 pallets of eggs, and put them back into the manual load zone, before the flour pallets would appear. Hope this helps.
So think of it like a greenhouse. The pickup area is more of just a visual display to show there is items being stored. When the pickup area is full everything else is stored inside the warehouse. What I did is put lettuce pallets into the warehouse until the pick up was full, kind of like a placeholder. Then I use the warehouse as normal and never move the lettuce pallets.