Using Falling Comet on a unit at a battlefield, then the other player plays Thrill of the Hunt, does Falling Comet still go through?

No, Falling Comet will not successfully deal damage to the unit in this scenario.

Explanation

According to RiftJudge FAQ #10035, when a unit is targeted by a spell and is then affected by an ability like Thrill of the Hunt that moves it to a non-board zone (Banishment), the targeting relationship is severed.

Sequence of Events

1. Chain Start: You play Falling Comet as an [Action] during a Showdown, targeting the enemy unit.
2. Reaction: Your opponent plays Thrill of the Hunt as a [Reaction] in response.
3. Chain Finalization: The chain is now: Thrill of the Hunt (top) and Falling Comet (bottom).
4. Resolution (LIFO):
   • Thrill of the Hunt resolves first: It banishes the unit and then plays it to a battlefield. Because the unit moved to a non-board zone, it ceases to be the "same game object" that was originally targeted. The targeting relationship is severed.
   • Falling Comet attempts to resolve: Because the specific unit it was targeting was moved to a non-board zone and is no longer the same game object, the targeting restriction is failed (mistargeted). As per Rule 359.3.e.7, the instruction to deal damage to that unit does not execute.