My opponent has a Stellacorn Hoarder on the battlefield. Can he use Ride the Wind to instantly bring it to the other battlefield?

Yes, your opponent can use Ride the Wind to move their Stellacorn Herder from one battlefield to the other.

Here is the breakdown of why this is a legal play:

1. Card Usage: Ride the Wind is an [Action] card that allows the controller to "Move a friendly unit and ready it." As an [Action] card, it can be played during the owner's Main Phase (when the state is open) or during a Showdown.
2. Movement Mechanics: Per Rule 444.1, spells and abilities can cause units to move. Unlike standard movement, which is restricted by certain rules (like Rule 144.1.c), movement via spells like Ride the Wind is not bound by the same limitations, provided the card itself doesn't impose additional restrictions. Ride the Wind does not prohibit moving between battlefields.
3. Timing: If your opponent plays Ride the Wind while a Showdown is active (a common scenario for this interaction), it follows the standard chain rules. Once played, Ride the Wind is placed on the chain. If no other reactions are played, it resolves, and the Stellacorn Herder immediately moves to the target battlefield and readies.
4. Trigger: Because Stellacorn Herder says "When I move, draw 1," this ability will trigger and be placed on the chain immediately after the unit arrives at the new battlefield, following the move's completion (Rule 441.3).

In summary: Yes, the move is legal. After the move, your opponent will draw a card from the Stellacorn Herder's triggered ability.