They can carry up to five stacks, but the drones precisely fulfill requests and thus are sent as soon as a few items are taken out of the delivery site. This is unexpected since logistics bots deliver their maximum carry size to fulfill requests(with the sole exception of the player's inventory). This is very inefficient in regards to drones used.

I can't figure out an easy way to prevent this, since you can't read a request depot's contents and also set its requests by circuit condition. I suppose I could use a latch circuit to set the request and then turn it off once the drone delivers the items, turning it back on once the depot is empty.

edit: well, the above concept doesn't quite work, since, again, I can't detect when the drone delivers items if I'm using the depot's circuit connection to request things.
Thankfully there's a workaround. You can read the contents of the entire logistic network via a roboport, and use THAT amount to detect when to request more of the item.