If an exhausted unit is moved to another battlefield after already having moved once, does it attack again?

Yes, an exhausted unit can enter combat if moved by a card effect, and it will become an attacker in that combat.

According to RiftJudge FAQ #8385, exhausted units can enter combat if they are moved there by card effects, as being exhausted only prevents regular (standard) moves, not movement by other methods.

When a unit is moved to a battlefield occupied by an opponent's units, it triggers a combat (or showdown) and gains the attacker designation. Because it is gaining the attacker designation for the first time in this new combat, it will trigger any "When I Attack" abilities it possesses, even if it had previously attacked in a different combat earlier that turn.

Key points to remember:

• Exhaustion State: Moving a unit via a card effect does not change its ready/exhausted state (unless the effect specifically says so).
• Attacker Designation: Per the Patch Notes (2025-10-24), "Attack and Defend triggers can happen once per combat, specifically the first time something gains the attacker or defender designation during that combat." Since this is a new combat, the unit gains the designation again and its triggers will activate.
• Standard Moves: You cannot perform a Standard Move with an already exhausted unit because exhausting the unit is the cost for that action (Rule 143.2). However, this restriction does not apply to moves caused by spells or abilities.