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Does Safeguard work if something tries to move a damage counter onto the character while it’s undamaged?

No. Moving damage counters doesn’t count as damaging the character, although after the damage counter is moved, the character will be damaged and Safeguard will no longer work. Sneaky!
Source: Wizards of the Coast (2019-11-08)    【Link to Ruling】
Tags: Safeguard