I believe all Fatalis run if the quest isn't a download quest. But they only run after you've dealt a good enough amount of damage to them taking out enough HP for them to run (maybe its 25% at least?). If you dont beat them by the 4th quest (reason why i said 25% because 4 x 25% = 100% health), i believe their HP gets reset. Also, downloaded Fatalis quest have lower HP than Guild Hall quests.
Like Locrin, i also bow Fatalis. I used a Peirce/Rapid and Dragon element bow to break his horns and scar the eye, chest(?), and wings. The Black and White Fatalis fight pretty much the same, except using a long range weapon against White Fatalis is a bit riskier since White Fatalis can call upon lightning attacks in an area. Also, i'm not 100% sure but i'm pretty sure Black Fatalis DOES NOT go into "Armor Mode" (a state where he turns all your damage to 10% of what it originally does). White Fatalis though, DOES have this mode once you've taken away 50% or so of his HP away. So some people recommend using weapons that cause explosion (Gunlances shells, Bowguns useing CragS and ClusterS) or using bombs against White Fatalis because those attacks can negate Fatalis's Armor Mode.
Also, even if you fail the quest (die 3x), the damage carries over for the Guild Hall Fatalis quest. So if you didn't kill or repel him but you fought him well for a good 20min or so, you might wanna consider saving and going after him again.
Good luck dawg.