The part of Ring of Thune that's important reads "At the beginning of your upkeep, put a +1/+1 counter on equipped creature if it's white".
If I was to move the ring after a couple turns, lose it to a Reclamation Sage or whatever, would the +1/+1 counters stay on the previously enhanced creature, or would that bonus leave with the ring?
The Ring of Thune puts counters on the creature, so the counters stay with the creature, not the equipment. If they wanted the power and toughness bonus to stay with the equipment instead of the creature, they would use a wording like on Sigil of Distinction:
Sigil of Distinction enters the battlefield with X charge counters on it.
Equipped creature gets +1/+1 for each charge counter on Sigil of Distinction.
"At the beginning of your upkeep, put a +1/+1 counter on equipped creature if it's white" is a triggered ability of the artifact "Ring of Thune".
It triggers at the beginning of your upkeep, if Ring of Thune is on the battlefield, and it references "equipped creature". If there is no creature currently equipped with Ring of Thune, the ability does nothing. If there is, that and only that creature get a +1/+1 counter.
The +1/+1 counters stay on the creature, they don't belong to the equipment.
