Heavy Stroke:

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Response: After a Dwarf deals X damage to an enemy during combat, deal and additional X damage to that enemy. (Limit once per phase.)

Azain Silverbeard:

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Response: After Azain Silverbeard participates in an attack that destroys an enemy, spend 1 Tactics resource to deal 2 damage to another enemy that shares a Trait with the destroyed enemy. Any player may trigger this response.

Can the damage dealt by Azain's Response be doubled via Heavy Stroke? I'm inclined to say no, but I am curious to see what others think.

The question would be whether the damage dealt by Azain's response is actually dealt by Azain. My guess would be that his direct damage, and all other direct damage effects (e.g. Thalin, Gandalf, Argalad, Farmer Maggot) is dealt by the player, not the character.

Agreed, Heavy Stroke wouldn't work here.