Does Bestow Curse trigger the Grim Harvest?

If I cast Bestow Curse on a monster then shoot it with a crossbow, would it trigger the Grim Harvest ability due to the extra d8 of damage? Does it matter if the monster had 1 hp?

What about if I cast Bestow Curse then hit it with Magic Missile, would Grim Harvest trigger off the missile or the curse?