If I play Wages of Pain and my opponent plays a Ride the Wind and moves the unit to another battlefield, does the Wages fizzle?

No, Wages of Pain does not fizzle. It still deals 3 damage to the unit after it moves.

Here's the sequence:

1. Wages of Pain goes on the Chain targeting the unit.
2. Opponent responds with Ride the Wind — it goes above Wages (LIFO).
3. Ride the Wind resolves first, moving that unit to another battlefield and readying it.
4. Wages of Pain then resolves. The unit is still "a unit at a battlefield" — it just changed which battlefield — so it remains a legal target. The 3 damage is dealt.
5. The Gold token from Wages is created at the original battlefield, not the new one (FAQ #8451).

If the unit survives the 3 damage, a new showdown starts at the destination battlefield where it landed (FAQ #7596).