If an unempowered Akali, Deadly Weapon moves in and out of a battlefield held by a Master Yi's 2 might unit, will the 2 might unit die?

It depends on how she leaves.

Master Yi's +2 only turns on once the unit holds the Defender designation, which is granted when the Combat Showdown opens — after the Chain empties (Rule 464.2.c.3). Akali's move damage always resolves before that, so at 2 Might she's dealing into a plain 2-Might body.

If she's pulled out at Reaction speed (e.g. Flash in response to her own move trigger), no Showdown ever opens — Contested drops because she controls no units there — so both 1-damage triggers resolve at 2 Might, 2 damage is lethal, and there's no Combat Cleanup to heal it: yes, the unit dies (FAQ #11658).

If she's pulled out with the Rogue Assassin legend instead, the Showdown has already started by the time you can activate it (it's an [Action] ability), so the lone unit is the Defender with Yi's +2 — 4 Might with 1 damage marked. Akali's second point of damage makes 2 of 4, not lethal, and Combat Cleanup heals it: it survives.