I contest an uncontrolled battlefield with Yone (5 might), my opponent rides the winds a 2 might unit in, then uses their focus to rune prison my Yone. I relentless pursuit my Yone to the other uncontrolled battlefield, and my opponent follows with ride the winds. What is the resolution on both battlefields?

To resolve this situation, we must look at how battlefields, contested status, and movement interact.

Battlefield 1: Because you moved Yone away using Relentless Pursuit and your opponent moved their unit away using Ride the Wind, there are no units left on Battlefield 1. It becomes Uncontrolled.
Battlefield 2: A new showdown begins here because your move to this battlefield made it Contested.

1. Battlefield 1 (Initial State)

You move Yone to an uncontrolled battlefield, making it Contested.
Your opponent moves a 2-might unit there using Ride the Wind. The battlefield remains Contested (Rule 184.3.b).
Your opponent uses Rune Prison on Yone. Yone is now Stunned (cannot deal combat damage this turn).

2. The Move to Battlefield 2

You use Relentless Pursuit to move Yone to the second uncontrolled battlefield.
Your opponent uses Ride the Wind to move their unit to the second battlefield as well.

3. Resolution of Battlefield 1

According to Rule 444.2.d: "If there are no Units remaining here controlled by any player, the Battlefield becomes Uncontrolled."
Since both units have left, Battlefield 1 is now Uncontrolled. No one conquers it, and no points are scored.

4. Resolution of Battlefield 2

When you moved Yone to the second battlefield, it became Contested (Rule 184.3.a).
When your opponent moved their unit there, it remained Contested.
A Staged Combat (Rule 345.2.b) will occur here.
Attacker/Defender: You applied the Contested status first (by moving Yone there), so you are the Attacker (Rule 345.2.b.1). Your opponent is the Defender.
Combat Outcome: Because Yone is currently Stunned (from the Rune Prison played earlier), Yone will not deal combat damage during this showdown. If your opponent's unit has 2 might, they will deal 2 damage to Yone. If Yone survives, control will be determined at the end of the showdown (Rule 184.4).

Stun: Rune Prison stuns the unit. This effect persists even if the unit moves to a different battlefield, as it is a status applied to the unit for the turn.
Conquer: You only "Conquer" a battlefield if you establish control at the end of a showdown (Rule 444.2.b). Since you are currently in a Staged Combat on Battlefield 2, you cannot Conquer it until that combat resolves and you have units remaining there.
Yone's Ability: Yone's ability ("When I conquer a battlefield that was uncontrolled...") will only trigger if you successfully win the showdown on Battlefield 2 and establish control, provided you have not already scored that battlefield this turn.