  An investigation of the isospin equilibration process in the reactions
$^{58,64}$Ni+$^{58,64}$Ni at two bombarding energies in the Fermi regime
($32\,$MeV/nucleon and $52\,$MeV/nucleon) is presented. Data have been acquired
during the first experimental campaign of the coupled INDRA-FAZIA apparatus in
GANIL. Selecting from peripheral to semi-central collisions, both the neutron
content of the quasiprojectile residue and that of the light ejectiles coming
from the quasiprojectile evaporation have been used as probes of the dynamical
process of isospin diffusion between projectile and target for the asymmetric
systems. The isospin transport ratio technique has been employed. The
relaxation of the initial isospin imbalance with increasing centrality has been
clearly evidenced. The isospin equilibration appears stronger for the reactions
at $32\,$MeV/nucleon, as expected due to the longer projectile-target
interaction time than at $52\,$MeV/nucleon. Coherent indications of isospin
equilibration come from the quasiprojectile residue characteristics and from
particles ascribed to the quasiprojectile decay.
