  The discrepancy between the CDF measurement and the Standard Model
theoretical prediction for the $W$-boson mass underscores the importance of
conducting high-precision studies on the $W$ boson, which is one of the
predominant objectives of proposed future $e^+e^-$ colliders. We investigate in
detail the production of $W$-boson pairs at $e^+e^-$ colliders, and compute the
next-to-next-to-leading order mixed QCD-EW corrections to both the integrated
cross section and various kinematic distributions. By employing the method of
differential equations, we analytically calculate the two-loop master integrals
for the mixed QCD-EW virtual corrections to $e^+e^- \rightarrow W^+W^-$.
Utilizing the Magnus transformation, we derive a set of canonical master
integrals for each integral family. This canonical basis satisfies a system of
differential equations in which the dependence on the dimensional regulator is
linearly factorized from the kinematics. We then express all these canonical
master integrals as Taylor series in $\epsilon$ up to $\epsilon^4$, with
coefficients articulated in terms of Goncharov polylogarithms up to weight
four. Upon applying our analytic expressions of these master integrals to the
phenomenological analysis of $W$-pair production, we observe that the
$\mathcal{O}(\alpha\alpha_s)$ corrections are significantly impactful in the
$\alpha(0)$ scheme, particularly in certain phase-space regions. However, these
mixed QCD-EW corrections can be heavily suppressed by adopting the $G_{\mu}$
scheme.
