If an attack causes the pokemon that has just used Destiny Bond to faint, the opposing pokemon will faint as well. The effects of Destiny Bond wear off as soon as the using pokemon's next turn begins, regardless of when they acted in the round when Destiny Bond was used.
If what causes the pokemon that used Destiny Bond to faint does not directly come from an attack used by an opposing pokemon, then Destiny Bond will not cause the opposing pokemon to faint. Examples of this would be damage from poisoning, Leech Seed, Sandstorm, as well as the effects of Perish Song. If the damage from Future Sight causes a pokemon using Destiny Bond to faint and the pokemon that initiated the attack has either switched out or already fainted, the current opposing pokemon will be finted by Destiny Bond.
Destiny Bond is not subject to the effects of Accuracy or Evade modifiers.