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What if your opponent puts those 3 damage counters on a character with 10 health and 9 damage counters already on it? Do the extra damage counters on go to a third character?

No. The “Steamroll damage” only carries over to one other character. That 10-health character would end up in the KO area with 10 damage counters on it. The extra 2 damage counters are lost.
Source: Wizards of the Coast (2019-03-01)    【Link to Ruling】
Tags: Steamroll