If I am defending with Kayn, can I play two Ride the Winds (or 1 Flash and 1 Ride the Wind) to move elsewhere and back to proc the invulnerability? Or does the showdown immediately end and I lose control the moment Kayn leaves the battlefield?

You can perform that sequence, and the showdown will not end. You will maintain control of the battlefield, and Kayn will gain his invulnerability effect.

According to RiftJudge FAQ #5510, when a unit leaves the battlefield during combat, the showdown does not end. You are permitted to move Kayn away and bring him back to the battlefield during the same showdown.

Why this works:

1. Showdown Continuity: The showdown persists as long as the state remains contested. Even if your battlefield becomes empty because Kayn moves to your base, the combat showdown remains ongoing.
2. Control: Per Rule 187.4.c. (updated in the Unleashed rules), you do not lose control of a battlefield if an item (like a spell) is on the chain or if a combat/showdown is ongoing there.
3. Kayn's Ability: Kayn's ability states: "If I have moved twice this turn, I don't take damage." Moving him via Ride the Wind (or Flash then Ride the Wind) counts as movement. As long as you complete two movements, the invulnerability condition is met.

Example Sequence:

• Opponent initiates showdown: Kayn is the defender.
• Move 1: Use Flash (or the first Ride the Wind) to move Kayn to your base. Combat continues despite there being no units on the battlefield.
• Move 2: Once you have focus, use Ride the Wind to move Kayn back to the battlefield. He arrives readied.
• Result: Kayn has now moved twice this turn. He is on the battlefield as the defender and is invulnerable to damage for the remainder of the showdown.

Because Kayn cannot take damage, if the attacking units fail to defeat him through other means (like movement effects or non-damage killing abilities), the attackers will be recalled to base at the end of combat, and you will succeed as the defender.

Disclaimer: This is my interpretation based on the specific interaction of mechanics defined in RiftJudge FAQ #5510 and the updated Rule 187.4.c. No official FAQ exists for this exact two-card combo, but it follows the established rulings for Kayn and combat movement.