If Duplicant exiles two cards due to Strionic Resonator, it uses the last one exiled to determine the P/T of Duplicant.

What happens if the last card exiled leaves exile, say by being processed?

Does the first card exiled become the de facto last card exiled?
Or is the object that it is referencing no longer in existence and as a result the link is broken?

Duplicant's static ability looks at the set of cards in the exile zone that were put there by its ETB trigger, and references the creature card currently within that set that's most recently been exiled. If the contents of the set change in any way, that may change which which creature card among it's most recently been exiled, and Duplicant's creature types and P/T will therefore change accordingly.