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We consider the notion of the graph product of actions of groups$\left\{G_v\right\}$ on a $C^*$-algebra $\mathcal{A}$ and show that undersuitable commutativity conditions the graph product action $\bigstar_\Gamma\alpha_v: \bigstar_\Gamma G_v \curvearrowright \mathcal{A}$ has the Haagerupproperty if each action $\alpha_v: G_v \curvearrowright \mathcal{A}$ possessesthe Haagerup property. This generalizes the known results on graph products ofgroups with the Haagerup property. To accomplish this, we introduce the graphproduct of multipliers associated to the actions and show that the graphproduct of positive definite multipliers is positive definite. These resultshave impacts on left transformation groupoids and give an alternative proof ofa known result for coarse embeddability. We also record a cohomologicalcharacterization of the Haagerup property for group actions.

On graph products of multipliers and the Haagerup property for $C^*$-dynamical systems