If I first use Bellow Breath and then play Watcher, will an enemy unit of 4 might die? How does the process work?

Yes, an enemy unit with 4 Might will die in this scenario.

According to RiftJudge FAQ #9019 and #10231, a unit is killed if it has marked damage equal to or greater than its current Might during a lethal damage check (which occurs after an effect resolves). Because damage remains marked on a unit and does not reduce its Might, lowering a unit's Might while it is already damaged can make that damage lethal.

Sequence of Events

1. Play Bellows Breath: You play Bellows Breath. As the spell resolves, you deal 1 damage to the enemy unit. This damage is "marked" on the unit.
2. Play Thousand-Tailed Watcher: You play the Thousand-Tailed Watcher. Its "When you play me" triggered ability is placed on the chain.
3. Resolve Watcher's Ability: As the ability resolves, it applies -3 Might to enemy units. The enemy unit's Might is reduced from 4 to 1.
4. Lethal Check: After the ability resolves, the game checks the unit's status. The unit now has 1 damage marked on it and has 1 Might. Since the marked damage (1) equals the unit's current Might (1), the unit is killed.

Key Rules

• Damage vs. Might: Damage is "marked" on a unit and does not reduce its Might value. A unit dies only when marked damage meets or exceeds its current Might (Rule 323.5.3b).
• Resolution: Because the Watcher is played after the Bellows Breath damage is already marked, the unit's Might reduction makes the existing damage lethal.
• Timing: As established in RiftJudge FAQ #3100, you cannot prevent the Thousand-Tailed Watcher's ability from resolving by killing the unit in response; the ability will still resolve even if the Watcher is removed from the board.