  We investigate the density behaviour of the symmetry energy with respect to
isospin equilibration in the combined systems $Ru(Zr)+Zr(Ru)$ at relativistic
energies of 0.4 and $1.528 AGeV$. The study is performed within a relativistic
framework and the contribution of the iso-vector, scalar $\delta$ field to the
symmetry energy and the isospin dynamics is particularly explored. We find that
the isospin mixing depends on the symmetry energy and a stiff behaviour leads
to more transparency. The results are also nicely sensitive to the "fine
structure" of the symmetry energy, i.e. to the covariant properties of the
isovector meson fields. The isospin tracing appears much less dependent on the
in-medium neutron-proton cross-sections ($\sigma_{np}$) and this makes such
observable very peculiar for the study of the isovector part of the nuclear
equation of state. Within such a framework, comparisons with experiments
support the introduction of the $\delta$ meson in the description of the
iso-vector equation of state.
