Question: Digitalis is a gray mesh that carries Creeper really fast. You can't destroy it, an in some maps, you can't avoid building on it. Whatever you build on it is prone to getting destroyed quickly whenever your defenses fall a bit behind, because then the Creeper rushes across the Digitalis quickly and destroys your Relays, and it's all downhill after that.

I've noticed that, after I destroy an Emitter, sometimes *all of* the Creeper that's in the Digitalis will "fall out" onto the ground and spread around like the viscous liquid it is. (Sometimes it's good to not have Creeper in the Digitalis, and sometimes it's a right pain to have it splashing around.) But I've also seen this happen a bit randomly, when I'm shooting Digitalis.

So exactly what circumstances does it take to make the Creeper fall out of the Digitalis?

Answer: 
The scenario you are describing occurs when the charged Digitalis is isolated from its source. The source of each group of Digitalis must be an Emitter that is located ON the Digitalis itself - an Emitter near a section of Digitalis does not charge it.

Isolating Digitalis means one of two things, either destroying its source, or cutting off a group of charged Digitalis from the source - by destroying some Digitalis in the middle of a charged Digitalis pathway.

It is useful to keep this second method in mind; I have often accidentally isolated a group of charged Digitalis within my base (because I thought it was of no further threat) to my impending doom and/or destruction. 

I have just double-checked all of this on a map I created for the purpose.
There are two kinds of Digitalis. I can't exactly recall the terms, so I will refer to them as "destroyed" and "constructed" Digitalis. Most/all of the Digitalis on the map starts in the destroyed state when a new map is started, although a few maps don't follow that. Destroyed Digitalis will absorb a little Creeper when they come in contact, destroying the Creeper and "constructing" the Digitalis.

Constructed Digitalis works like you say - it picks up and transports Creeper, making it a great way for Creeper to attack you up a hill or across an empty space. However, attacks by your weapons can destroy the Digitalis again, reverting it to its inactive state. This will stop it from carrying Creeper, so damaging the Digitalis while there's a lot of Creeper being carrying by it will make it drop and spread normally. Given a little time with the Creeper, it'll absorb a bit of Creeper again and repair itself once more.