Update
Starting from 0.16 it seems it will be possible to use a new type of buffer chest explicitly for this purpose: https://www.factorio.com/blog/post/fff-203
I have built several variations of the circuit-based approach described below since I wrote this answer to improve the design and work out annoyances (like no items ending up in the storage chest until requests have been satisfied). None of them have worked perfectly yet or became so complex they can't easily be explained so therefore I have not updated this answer yet. With a future (as of this writing) 0.16.x release it seems this problem will go away completely.
The new buffer chests (green) will be a hybrid between requester chests (blue) and passive provider chests (red). I presume the robots will attempt to fill these new chests before putting anything into generic storage chests (yellow), and also grab from them before going to storage or passive providers, given the images and descriptions in that blog post:
By using a buffer chest, you can setup a dedicated 'supply area', where the buffer chest will already contain all the typical items, and the bots can quickly top-up your inventory.
...
Using the buffer chest, it will be easy to setup nearby supplies to
quickly repair the walls when needed.
- Factorio devs
Circuit network approach
The idea is to prevent an infinite loop by not requesting anything when it's not needed, and not allowing anything to be picked up from your remote depot until it has been filled completely, at which point it makes everything available to the player (or any other requester chests nearby).
- Put a requester chest somewhere near where you want things delivered,
do not make it request anything yet. Make sure it's set to "set requests".
- Put a regular non-logistics chest next to it, leaving one space between them. This will be our "buffer" to prevent loops.
- Put an inserter between the chests, rotated so it picks from the requester chest and places into the regular chest.
- Put a storage chest 1 tile away from the regular chest.
- Put an inserter between those chests so it takes from the regular chest and puts it into the storage chest.
- Put a decider combinator somewhere nearby, set it to trigger if combat supply X goes below Y amount. Set the output type to "1 of X".
- Grab a wire (say red) and connect the regular and storage chests together with the inserter which takes from the regular chest, and connect them with the input side of the combinator. The combinator should now show the current total amount of supply X in the depot as its input value, and the output value should be 1 of that type.
- Set the condition on that inserter to only trigger if supply X is equal to the amount Y.
- Connect the output of the combinator to the requester chest.
The depot should now slowly begin to fill the regular "buffer" chest via the requester chest and the first inserter between them. Once the target amount Y has been reached the requests will stop and the other inserter will take over and dump everything into storage. Requests will resume once the total number of X-items in the chests goes below the Y threshold.
Since bots prioritize fetching from storage chests, they will likely grab from there over a passive provider chest further away when you pass by.
If there's stuff left in the storage chest after you're tanked up, they will be moved back to the requester chest by the bots and end up in the buffer chest again, until the're eventually dumped back into the storage chest when the system has reached the threshold.
There are ways to tweak this system for various purposes and uses.
If you want to fill the depot faster; put an arithmetic combinator between the first combinator and the requester chest, and set it to multiply the input amount (1) by some factor (say 10) (make sure to output the same type). The chest will keep requesting 10 items until the system is full (prepare for some overflow as bots always grab/deliver a full cargo hold if they can).
You could use more combinators to always request the exact number of items left until the system is full, but simply multiplying the number of items requested uses fewer components.
If bots often end up re-filling the system from itself by emptying lots of items from the storage chest into the requester, add another decider combinator (using the green network) which triggers if the storage chest contains Z items, where Z is much lower than the total amount above (Y) and set it so the inserter only grabs that amount from the buffer.
It may take slightly longer to restock the player as the storage chest is emptied out more often, but the bots will more likely restock the depot more efficiently as there's not as much "available" from the depot itself.
You could also let the filling of the storage chest be player-controlled by hooking up a state-reading gate pole (or even multiple ones at different locations) to the iserter and only allowing it to operate while you're standing near the gate(s). But if you have to run to a specific location you may as well grab directly from the buffer chest, perhaps conveniently placed near your gates.
Best Answer
The setup in the OP's linked image works something like this:
The requester chest will sum all its circuit inputs and request exactly that. The left combinator sends out 1 copper plate as long as the roboport reports there is less than some amount available in the logistics system.
The right combinator will add -1 copper plate to the sum requested (totalling to 0), but only when the provider chest has anything in it.
Basically that system will request 1, immediately move it to the provider, which one tick later adds -1 to the requested amount. Once something empties the provider the requester will once again request 1.
The inserter "overfills" the requester even if it's not requesting anything, as long as the total amount of plates is below 200 in the logistics network (NOT the circuit network), to allow a larger buffer than just 1 item at a time. (Note: Depending on your bot and inserter upgrades, you may never get exactly the number of items you request, but you will never get less than that in one delivery.)
The key here is that there is currently no direct way to automate switching of chest types or stop providing something, but requester chests can be told to only request items when a circuit condition is met.
Another important thing to note is that fetching objects from Storage chests take priority over fetching from Passive Provider chests. The setup in the image does not use this, but it can be used to prioritize where robots fetch from.
This is why I sometimes prefer the slightly different approach I describe in my answer here: https://gaming.stackexchange.com/a/292661/74907.
This question seems limited to using the logistics network only, where the above mentioned question could be interpreted as allowing any approach to obtaining the goal of moving useful items closer to some specific location on the map. I guess they overlap but aren't exact duplicates.
Rather than repeating my entire answer from the other question here, I'll just say that the key is to move all items from the requesting chest into a [regular type] buffer-chest for temporary storage until the total amount of items at the location is enough to stop requesting. Once that condition is met and requesting has stopped, all items can be moved to a Storage chest. Robots will the prioritize fetching from that chest even if there are other sources nearby.
Finally you need a condition at which to begin requesting again, at which point your storage chest will likely first be drained into the requester chest (you could speed that up by also enabling an inserter between them). That will not cause a loop as once the storage chest is empty and the requester chest says it wants more items, they will be gathered from elsewhere (this is why your buffer chest must not be another requester, storage or provider).
I've got storage chests scattered all over the place so my bots can drop overflow off anywhere practical for them, and I only user passive provider chests as outputs from factories, which become "inputs" to the logistics network(s). As long as bots can get what they need from storage I won't waste energy producing more of those items. If the factory is large and the bots get tempted to ignore my storage and get things from passive providers anyway, I've set the inserters emptying my factories to not empty while the logistics network still has more than X amount in it. The the bots will be forced to drain the storage chests and reduce the amount of over-production.
Just make sure the bots can actually get what they need within each logistics network...