If the Bestow Curse spell causes me to do extra damage, does the target’s death trigger the Necromancy wizard’s Grim Harvest feature?

In D&D 5e using a wizard of the School of Necromancy:

If I cast bestow curse on a monster, then kill it with a crossbow, would it trigger the Grim Harvest feature due to the extra 1d8 damage? Does it matter if the monster had 1 HP?

What about if I cast bestow curse, then hit it with magic missile (and kill it with just 1 missile) – would Grim Harvest trigger off the missile or the curse?