If Yasuo, Remorseful attacks a battlefield, deals 6 damage to a unit with his ability, then is moved away with Ride the Wind, does the damaged unit heal or does the damage stick?

The damage will heal during Combat Cleanup. Here's the sequence: (1) Yasuo attacks and his ability triggers, dealing 6 damage to an enemy unit at the battlefield. (2) Ride the Wind moves Yasuo away and readies him, but the 6 damage remains marked on the unit. (3) The damage stays on the unit throughout combat. (4) During Combat Cleanup , all units heal. (5) The 6 damage is removed from the unit. So while the damage sticks temporarily through combat, it will automatically be healed during the Combat Cleanup phase that occurs at the end of combat (assuming the unit survives and the 6 damage doesn't kill it before cleanup).