The notion of disjoint $\mathcal{A}$-transitivity for a Furstenberg family $\mathcal{A}$ is introduced with the aim to generalize properties derived from disjoint hypercyclic operators. We begin a systematic study by showing some of the basic properties, including necessary conditions to inherit the property on the whole space from an invariant linearly dense set containing the origin. As a consequence, we continue the study of the link between non-linear and linear dynamics through Lipschitz-free spaces by presenting some necessary conditions to obtain disjoint $\mathcal{A}$-transitivity for families of Lipschitz-free operators on $\mathcal{F}(M)$ expressed in terms of conditions in the underlying metric space $M$.
