  We propose a transport theory for the kinetic evolution of solar-wind
electrons in the heliosphere. We derive a gyro-averaged kinetic transport
equation that accounts for the spherical expansion of the solar wind and the
geometry of the Parker-spiral magnetic field. To solve our three-dimensional
kinetic equation, we develop a mathematical approach that combines the
Crank--Nicolson scheme in velocity space and a finite-difference Euler scheme
in configuration space. We initialize our model with isotropic electron
distribution functions and calculate the kinetic expansion at heliocentric
distances from 5 to 20 solar radii. In our kinetic model, the electrons evolve
mainly through the combination of the ballistic particle streaming, the
magnetic mirror force, and the electric field. By applying fits to our
numerical results, we quantify the parameters of the electron strahl and core
part of the electron velocity distributions. The strahl fit parameters show
that the density of the electron strahl is around 7% of the total electron
density at a distance of 20 solar radii, the strahl bulk velocity and strahl
temperature parallel to the background magnetic field stay approximately
constant beyond a distance of 15 solar radii, and $\beta_{\parallel s}$ (i.e.,
the ratio between strahl parallel thermal pressure to the magnetic pressure) is
approximately constant with heliocentric distance at a value of about 0.02. We
compare our results with data measured by Parker Solar Probe. Furthermore, we
provide theoretical evidence that the electron strahl is not scattered by the
oblique fast-magnetosonic/whistler instability in the near-Sun environment.
