  We extend the microscopic Fermi-liquid theory for the Anderson impurity
[Phys.\ Rev.\ B {\bf 64}, 153305 (2001)] to explore non-equilibrium transport
at finite magnetic fields. Using the Ward identities in the Keldysh formalism
with the analytic and anti-symmetric properties of the vertex function, the
spin-dependent Fermi-liquid corrections of order $T^2$ and $(eV)^2$ are
determined at low temperatures $T$ and low bias voltages $eV$. Away from
half-filling, these corrections can be expressed in terms of the linear and
non-linear static susceptibilities which represent the two-body and three-body
fluctuations, respectively. We calculate the non-linear susceptibilities using
the numerical renormalization group, to explore the differential conductance
$dI/dV$ through a quantum dot. We find that the two-body fluctuations dominate
the corrections in the Kondo regime at zero magnetic field. The contribution of
the three-body fluctuations become significant far away from half-filling,
especially in the valence-fluctuation regime and empty-orbital regimes. In
finite magnetic fields, the three-body contributions become comparable to the
two-body contributions, and play an essential role in the splitting of the
zero-bias conductance peak occurring at a magnetic field of the order of the
Kondo energy scale. We also apply our microscopic formulation to the
magneto-resistance and thermal conductivity of dilute magnetic alloys away from
half-filling.
