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If you manage to put an opponents card form exile into their graveyard what ever ability that put it there originally will not be track. So for Identity Thief, when you attack and exile their creature you can then use Oracle of Dust to put it into their graveyard essentially killing the creature. For the Brisella combo this won't work because the cards exile and return immediately so there is no chance for you to put them in the graveyard. This works for anything that temporally puts them into exile.
When it comes to an effect like melding Brisela, where the cards are exiled an immediately returned from the same ability, there isn't any interaction you can have with the cards in exile, since you don't get priority during the resolution of a spell or ability.
However, if the card stays in exile long enough for you to get priority, you can certainly process it or target it with Pull from Eternity or whatever. What that means exactly depends on the card that exiled it. Cards like Identity Thief or Banishing Light won't return the card from the graveyard to the battlefield. Duplicant will revert to its original power and toughness. And so on.
Identity Thief will cause both cards to be exiled (and it becomes a copy of Brisela, Voice of Nightmares). As soon as they leave the battlefield, the two cards become new objects, separate and front face up. They would both return at the beginning of the next end step, but not melded. While they are exiled, Oracle of Dust can "grab" one of them per activation. Any of them that gets put in the graveyard this way won't return with Thief's trigger, because the move from exile to graveyard makes it once again a new object that Thief doesn't track, it can only return the cards it put in exile, from exile.
EDIT : I read the question as the scenario where Brisela would be exiled by Thief, the above posters are correct that you cannot stop the Meld process with something like Oracle of Dust.
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I'm a former judge (lapsed), who keeps up to date on rules and policy. Keep in mind that judges' answers aren't necessarily more valid than those of people who aren't judges; what matters is we can quote the rules to back up our answers. When in doubt, ask for such quotes.
If Identity Thief exiles Bruna or Gisela, it won't return until after the time Gisela's ability would have triggered that turn. The ability might trigger on your next turn, though.
513. End Step
513.2. If a permanent with an ability that triggers “at the beginning of the end step” enters the
battlefield during this step, that ability won’t trigger until the next turn’s end step. Likewise, if a
delayed triggered ability that triggers “at the beginning of the next end step” is created during this
step, that ability won’t trigger until the next turn’s end step. In other words, the step doesn’t “back
up” so those abilities can go on the stack. This rule applies only to triggered abilities; it doesn’t
apply to continuous effects whose durations say “until end of turn” or “this turn.” (See rule 514,
“Cleanup Step.”)
Oracle of Dust can't interfere with Gisela's meld ability.
608.2f If an effect gives a player the option to pay mana, he or she may activate mana abilities
before taking that action. If an effect specifically instructs or allows a player to cast a spell
during resolution, he or she does so by following the steps in rules 601.2a–i, except no player
receives priority after it’s cast. That spell becomes the topmost object on the stack, and the
currently resolving spell or ability continues to resolve, which may include casting other spells
this way. No other spells can normally be cast and no other abilities can normally be activated
during resolution.
701.35. Meld
701.35a Meld is a keyword action that appears in an ability on one card in a meld pair. (See rule
712, “Meld Cards.”) To meld the two cards in a meld pair, put them onto the battlefield with
their back faces up and combined. The resulting permanent is a single object represented by two
cards.
Although, Oracle of Dust can keep Identity Thief from returning a card to the battlefield.
400.7. An object that moves from one zone to another becomes a new object with no memory of, or
relation to, its previous existence. There are eight exceptions to this rule:
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Why bother with mere rulings when so many answers can be found in the Rules?
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However, if the card stays in exile long enough for you to get priority, you can certainly process it or target it with Pull from Eternity or whatever. What that means exactly depends on the card that exiled it. Cards like Identity Thief or Banishing Light won't return the card from the graveyard to the battlefield. Duplicant will revert to its original power and toughness. And so on.
Two Score, Minus Two or: A Stargate Tail
(Image by totallynotabrony)
EDIT : I read the question as the scenario where Brisela would be exiled by Thief, the above posters are correct that you cannot stop the Meld process with something like Oracle of Dust.
Oracle of Dust can't interfere with Gisela's meld ability.
Although, Oracle of Dust can keep Identity Thief from returning a card to the battlefield.