  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 $M$.
