Does Ride the Wind allow you to move between battlefields or just back and forth between the base and battlefields?

Yes, Ride the Wind allows you to move a unit from one battlefield to another.

According to multiple RiftJudge FAQs (such as FAQ #7631, #4995, and #1188), the restriction on moving between battlefields applies only to the Standard Move. Because Ride the Wind is a spell effect rather than a Standard Move, it is not restricted by the rules regarding battlefield-to-battlefield movement.

As stated in FAQ #1206: "Cards like Charm and Ride the Wind have no restrictions on unit movement between battlefields... These cards do not force units to perform their Standard Move Action; they simply move units."