For anyone looking for info on how the new Chaos and Nagash Crisis' work, here are my shorthand notes on various functions.
I didn't bother with the Skaven one because the AI seems to easily end this crisis without player intervention. The AI struggles super hard against the Chaos armies, but they absolutely destroy the Skaven. Let me know if I got anything wrong or if there's more to add.
20 turns before an invasion begins you get a warning. Then 10 turns prior you get another warning.
Chaos Invasions:
All chaos factions are destroyed and all chaos aligned legendary lords respawn with a crisis army in the chaos wastes along the north and southern edges of the world. All their former territory becomes devastated.
6 random areas on the map get an immediate invasion of 3 armies and a "Fortress" army. All of these areas immediately become devastated. Archaon spawns in one of these.
Beastmen armies respawn somewhere in the world.
All Norscan factions join Archaon's faction.
Archaon starts with 100% ward save. Destroying a stationary "fortress" army reduces that by 10%. Destroying an army reduces it by 1% per army. There are 50 armies in the world to hunt.
After defeating Archaon on the campaign map, you unlock a quest battle to truly defeat him and end the crisis.
Nagash Invasion:
10 turns before start all Mortarchs are removed from the world.
The Black Pyramid of Nagash and Nagashizzar return ownership to Nagash. Once the 20 turn warning timer starts, you can no longer attack these 2 settlements. And during the crisis you cannot attack them either.
All factions that are led by a Mortarch become owned by Nagash. All Mortarchs respawn at their respective capitals.
All remaining vampire factions spawn a bunch of armies at their capital and become vassals of Nagash. Dead factions respawn with a bunch of armies. These spawning armies are basically the same mechanic used for the old crisis.
Over 20 turns the Black Pyramid will return to Nagashizzar. During that time you have to kill armies to weaken his world destroying Ritual.
Defeating armies makes the final battle easier.
After 20 turns you play a quest battle to determine the fate of the world.