Tooltip on the relic is a little unclear but do these scale with player or pet bonuses? Do they pick up the flat damage from the Demo’ Flame Touched buff and/or from Hellfire Aura from the Hellhound? For that matter, the Total Speed buff from Manipulation? (They are damn slow)
Flame Touched and Hellfire Aura are both auras with a radius. That means they apply their bonus to any allies within that range, so your pets will get the bonuses from those unless they are excessively far away.
The bonuses from Manipulation do apply to Swarmlings.