Aligned Hedron Network and Ulamog's Despoiler
Ulamog's Despoiler makes no limitations as to what cards can be moved from exile to graveyard except the cards' owners. Therefore, you can move any of your opponent's exiled cards, no matter how they got exiled. Note, however, the rulings on the Despoiler's Gatherer page regarding face-down exiled cards.

If the Aligned Hedron Network was destroyed after Despoiler has had its effect, the Network's effect would end and return all cards exiled with it to the battlefield. Unless a few specific exceptions apply (which they don't here), effects do not track objects across zones, and the Network would not return the cards put into the graveyard by Despoiler.