  We study the ballistic-to-diffusive transition induced by the weak breaking
of integrability in a boundary-driven XXZ spin-chain. Studying the evolution of
the spin current density $\mathcal J^s$ as a function of the system size $L$,
we show that, accounting for boundary effects, the transition has a non-trivial
universal behavior close to the XX limit. It is controlled by the scattering
length $L^*\propto V^{-2}$, where $V$ is the strength of the integrability
breaking term. In the XXZ model, the interplay of interactions controls the
emergence of a transient "quasi-ballistic" regime at length scales much shorter
than $L^*$. This parametrically large regime is characterized by a strong
renormalization of the current which forbids a universal scaling, unlike the XX
model. Our results are based on Matrix Product Operator numerical simulations
and agree with perturbative analytical calculations.
