Loyalty cascades happen if anyone in your fortress sees two other people in the fortress fighting each other at higher than brawl levels. Apparently (and it seems to be the case based on my limited study) the dwarfs that witness the event will be split along family ties. So if one of the combatants is related to someone viewing the battle, then they will side with that combatant.
As an example, if dwarf A sees dwarf B and C fighting (above brawl levels), *and* if A is related to B, then A will side with B. It might also take friends into account, but I'm not sure. Because migrants tend to be related to other people in the fortress, if a loyalty cascade starts, then it can literally split the fortress into 2.
Note that loyalty cascades can be stopped easily simply by making sure that nobody *who is not already involved* can't see the event. Only dwarfs who can see the battle will get involved. Once they are involved, *you can't stop them*. Let them fight until one side is dead. It *really* helps to have lockable doors to reduce line of sight in your fortress.
The easiest way I've found to get a loyalty cascade by accident is to keep were-beasts in your fortress on purpose. When one of your dwarfs is a were-beast, *they are still seen as family to those watching*! So if someone sees someone attack a were-beast that belongs to your fortress, a loyalty cascade is likely.
Note that not all dwarfs who see the battle will join. The times I've checked, each combatant has been related to someone else in the battle (but I haven't checked that many times and since I know how to contain a loyalty cascade I don't really get more than 2-3 dwarfs involved at any one time). I have noticed that dwarfs with high discipline seem less likely to join the battle, but it may be coincidence.
The other main way I've seen loyalty cascades happen is when brawls in the tavern escalate. It's never happened in my fortresses but I've seen it more than once on Youtube play throughs.
I personally like loyalty cascades and consider them a great feature. The only thing I'd like to see is a way for a dwarf with high social levels to mediate the conflict. The fact that once a fight starts, it's to the death is pretty crap. Also, once a loyalty cascade has broken out once, it would be cool if the families involved were more prone to fighting each other...