  We study the van der Waals interaction between Rydberg alkali-metal atoms
with fine structure ($n^2L_j$; $L\leq 2$) and heteronuclear alkali-metal dimers
in the ground rovibrational state ($X^1\Sigma^+$; $v=0$, $J=0$). We compute the
associated $C_6$ dispersion coefficients of atom-molecule pairs involving
$^{133}$Cs and $^{85}$Rb atoms interacting with KRb, LiCs, LiRb, and RbCs
molecules. The obtained dispersion coefficients can be accurately fitted to a
state-dependent polynomial $O(n^7)$ over the range of principal quantum numbers
$40\leq n\leq 150$. For all atom-molecule pairs considered, Rydberg states
$n^2S_j$ and $n^2P_j$ result in attractive $1/R^6$ potentials. In contrast,
$n^2D_j$ states can give rise to repulsive potentials for specific
atom-molecule pairs. The interaction energy at the LeRoy distance approximately
scales as $n^{-5}$ for $n>40$. For intermediate values of $n\lesssim40$, both
repulsive and attractive interaction energies in the order of $ 10-100 \,\mu$K
can be achieved with specific atomic and molecular species. The accuracy of the
reported $C_6$ coefficients is limited by the quality of the atomic quantum
defects, with relative errors $\Delta C_6/C_6$ estimated to be no greater than
1\% on average.
