We extend the microscopic Fermi-liquid theory for the Anderson impurity [Phys. Rev. B 64, 153305 (2001)] to explore nonequilibrium transport at finite magnetic fields. Using the Ward identities in the Keldysh formalism with the analytic and antisymmetric properties of the vertex function, the spin-dependent Fermi-liquid corrections of order
T
2
and
(
e
V
)
2
are determined at low temperatures
T
and low bias voltages
e
V
. Away from half filling, these corrections can be expressed in terms of the linear and nonlinear static susceptibilities which represent the two-body and three-body fluctuations, respectively. We calculate the nonlinear susceptibilities using the numerical renormalization group, to explore the differential conductance
d
I
/
d
V
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 becomes 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 magnetoresistance and thermal conductivity of dilute magnetic alloys away from half filling.