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Is there a specific name for the plot device in which the story's "Big Bad" has a cunning plan to use some monstrous being to further their diabolical machinations. This may be something summoned from elsewhere, brought back from banishment after earlier issues, or even the creation of magic or mad science. But when it arrives/arises it turns around and destroys the existing Big Bad only to pursue its own campaign of destruction that continues to tie down the same protagonists?

Just Desserts

From TV Tropes:

A villain ultimately finds their evil deeds come back to bite them.
Literally—they end up getting eaten.

This does not include a Heroic Sacrifice. But may be subverted with a
minor character being killed and eaten in obvious foreshadowing of
what is going to happen to one of the bads at some point. While Mooks
may be recipients of the Just Desserts, a true Just Dessert is
reserved for those higher up the ladder. If the beast doing the eating
was unleashed by the guy who gets eaten, he's been Hoist by His Own
Petard, making him a Self-Disposing Villain. Bonus points if he made a
practice of feeding people to said beast beforehand.

On TV Tropes this is called Evil Is Not a Toy:

Sometimes the Sealed Evil in a Can doesn't escape by itself, nor is it released by an Unwitting Pawn, but is deliberately set free by a villain (or hero). Let's call him Bob. Bob usually thinks he can control the sealed evil, or bargain with it, expecting to trade on a certain level of gratitude on its part since he was the one who freed it (or in extreme cases, resurrected it).

This never works.

The Sealed Evil in a Can will inevitably turn on the one who freed it — sometimes sooner, sometimes later. In many cases, it turns out to have no understanding of loyalty or gratitude at all. Bob may end up being killed on the spot, or he may be enslaved by the sealed evil

[...]

In any case, if Bob was the Big Bad before, he was really just a Little Bad; the formerly-sealed evil is the true Big Bad.

There is also Eat The Summoner, which fits very closely to the title of this question:

Alice is a Sealed Evil in a Can. Bob, either because he was promised something like immortality or riches, thinks he can control her, or because he thinks Alice isn't as evil as everyone says she is, tries to free her. After a bit of hard work, and possibly some outmaneuvering of people with more common sense, Bob frees Alice. However, rather than rewarding Bob for his service or even giving him a simple thank you, Alice tosses Bob into her mouth and eats him without a second thought.

Personally from the little bit of research I've done I feel that this is originally the "Good vs. Evil" troupe but after the mega force or "worser" evil is unleashed who then attempts to destroy the original villain it switches troupes to what's called "Eviler Than Thou". This is what I believe would be the best way to explain it. Here's the article that lead me to believe that this was the best answer: Eviler Than Thou

I think Frankenstein fits this category almost perfectly. In Frankenstein the protagonist had created a creature that backfired on him and ruined his life. That pattern does not take into account that evil aspect as the protagonist wasn't evil. It just describes the pattern of creating something for the sake of personal benefit (be it a good or bad character) that backfires horribly. So maybe we can call it - "The Frankenstein Effect"?