They are similar in idea to European Wild Hunt legends, and are also occasionally called Wild Hunts.
Heaven has rules about when and how a Hunt can be declared and what happens next.
Because of heavenhounds' pack mentality, they make very organised and tireless hunters.
Angels hunt on foot or wing and carry weapons. They are not as swift as hounds on foot. The Heavenly Hunt is a test of skill and discipline for angels as well as hounds.
Hades also has rules, but these are argued about and renegotiated every time and usually end up being flouted by most of the participants.
Because each hellhound is bound to a demon and demons don't tend to trust each other, and also because demons can change shape to serve as 'steed', 'hound' or 'huntsman' (or any combination), a Hadean Hunt tends to be disorganised and chaotic. Demons don't need to carry weapons.
The quarry has a fairly high chance of escaping if it takes advantage of mishaps, infighting or breakdowns in communication. For this reason, several prey creatures are often used in a single Hunt.
Hunts may pass through any of the various worlds served by this hell and heaven, and some other worlds that aren't. If witnessed by the inhabitants of these worlds, they can give rise to rumours and legends.
There are stories about the occasional mortal being caught up in a Hunt (or managing to crash the party and join in). Because himmelhunde can inspire terror with their voices, and hellhounds are just damn scary, most sane people would not choose to do this.