  Although the general theory of relativity (GR) predicts that gravitational
waves (GWs) have exactly the same propagation velocity as electromagnetic (EM)
waves, many theories of gravity beyond GR expect otherwise. Accurate
measurement of the difference in their propagation speed, or a tight constraint
on it, could be crucial to validate or put limits on theories beyond GR. The
proposed future space-borne GW detectors are poised to detect a substantial
number of Galactic white dwarf binaries (GWDBs), which emit the GW as
semi-monochromatic signals. Concurrently, these GWDBs can also be identified as
optical variable sources. Here we proposed that allocating a GWDB's optical
light curve and contemporaneous GW signal can be used to trace the difference
between the velocity of GW and EM waves. Simulating GW and EM wave data from 14
verification binaries (VBs), our method constrains propagation-originated phase
differences, limiting the discrepancy between the speed of light ($c$) and GW
($c_{GW}$). Through the utilization of LISA's design sensitivity and the
current precision in optical observation on GWDB, our study reveals that a
four-year observation of the 14 recognized VBs results in a joint constraint
that confines $\Delta c/c$ ($\Delta c = c_{\mathrm{GW}} - c$) to the range of
$-2.1\times10^{-12}$ and $4.8\times10^{-12}$. Additionally, by incorporating
this constraint on $c_{\mathrm{GW}}$, we are able to establish boundaries for
the mass of the graviton, limiting it to
$m_{\mathrm{g}}\le3\times10^{-23}\,e\mathrm{V}/c^{2}$, and for the parameter
associated with local Lorentz violation, $\bar{s}_{00}$, constrained within the
range of $-3.4\times10^{-11}\le\bar{s}_{00}\le1.5\times10^{-11}$.
