  In collinear factorization, light-cone distribution amplitudes (LCDAs) are
key ingredients to calculate the production rate of a hadron in high energy
exclusive processes. For a doubly-heavy meson system (such as $B_c, J/\psi,
\Upsilon$ etc), the LCDAs contain perturbative scales that can be integrated
out and then are re-factorized into products of perturbatively calculable
distribution parts and non-relativistic QCD matrix elements. In this
re-factorization scheme, the LCDAs are known at next-to-leading order in the
strong coupling constant $\alpha_s$ and at leading order in the velocity
expansion. In this work, we calculate the ${\cal O}( { v}^2)$ corrections to
twist-2 LCDAs of S-wave $B_c$ mesons. These results are applicable to heavy
quarkonia like $\eta_{c,b}$, $J/\psi$ and $\Upsilon$ by setting $m_b=m_c$. We
apply these relativistically corrected LCDAs to study their inverse moments and
a few Gegenbauer moments which are important for phenomenological study. We
point out that the relativistic corrections are sizable, and comparable with
the next-to-leading order radiative corrections. These results for LCDAs are
useful in future theoretical analyses of the productions of heavy quarkonia and
$B_c$ mesons.
