Boguing
Think of an airship as a pint of Guinness. The darker part is akin to the envellope. The cream on top would be the ballonet.
The envellope holds the helium only. The ballonet(s) hold atmospheric air. The two are not connected in as far as transfering air to helium or helium to air. This would be terrible for purity (how 'good' the helium is) and other things. When we valve air from the ballonet, it gets dumped overboard. We would do this perhaps if we were climbing steeply and the envellope pressure is increasing to its upper limits. Valved helium also goes overboard, it is not re-cycled.
As we descend, the airship pressure decreases and we need to maintain this within AFM limits. We do this by letting atmospheric ram-air into the ballonet(s) to increase the pressure hence maintaining the shape of the structure. It's the same on the ground also, on a cold night it is possible for the ballonet(s) to approach their maximum inflation. Helium must be added immediately. This is why the FAA and CAA mandate a 24 hour watch system for ground crew when we don't fly.
A final danger would be valving too much helium in flight. It is theoretically possible to descend to lower heights and have a full ballonet... This means you cannot descend any further under present atmospheric conditions. Hurry up and wait!
To summate:
Climbing? Air is dumped from the ballonet overboard.
Flat ballonet in flight? pressure height. Need further climb? Dump helium overboard.
Descending? Allow air to enter the ballonet via ram-air scoops or fan assisted means.
Twin ballonets can be used like fixed wing elevator trim.
Temperature increases? Helium expands, ballonet vents air.
No ballonet left (pressure height) is not just a flight scenario, it can happen on the mast at the airport.
Greybeard: I've been overtaken by birds, r/c aircraft, shot at by Mongo (easier target than a Busch beer can) you name it... Enjoying my current type's 470 TAS FL 450 crz as a change!