It's combo attack that includes blm. "Moogle guard bla bla bla"
Every moogle has a solo skill, and a combo skill. Thus stacks on any moogle is bad bad bad.
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I don't think anyone has answered your question ><.
You die because the pld moogle applies a certain debuff on you which makes you take extra damage from the mrd moogle. Notice the red debuffs on the MT and OT next time you tank moogle again. It's basically an anti-add swap mechanic.
I'm surprised no one has mentioned this but the reason you would die so quickly is because the war and pld moogles give a debuff. If you have the pld debuff the warrior will hit you like a train and vice versa.
So pretty much like others have said, swap king.
Edit: Ninja'd!
You only swap the king back and forth and you can switch at 2 stacks, makes it easier to heal then doing 3.