Can Irelia move to another battlefield using Ride the Wind while she has Ganking from Swiftness Boots, and if so, does a showdown start and can a point be earned on the opponent's turn?

Ruling: Yes, Irelia can move to another battlefield with Ride the Wind regardless of Ganking status. A showdown will be staged at the destination battlefield once the game is in a neutral open state (no other showdown ongoing and no active chain), and you can score points on the opponent's turn.

Sequence:

• Ride the Wind moves Irelia to the target battlefield
• Once the game reaches a neutral open state, a showdown is staged at that battlefield
• Points can be scored during this showdown even though it's the opponent's turn

Nuances:

• Movement abilities like Ride the Wind ignore Ganking and ready/exhausted restrictions, which only apply to the Standard Move Action
• Vilemaw's Lair only restricts moving from the battlefield to base, not to other battlefields
• Ride the Wind cannot move a unit to base from Vilemaw's Lair because that restriction does apply to all moves