  We perform a systematic study of merging black hole (BH) binaries with
compact star (CS) companions, including black hole--white dwarf (BH--WD), black
hole--neutron star (BH--NS) and black hole--black hole (BH--BH) systems.
Previous studies have shown that mass transfer stability and common envelope
evolution can significantly affect the formation of merging BH--CS binaries
through isolated binary evolution. With detailed binary evolution simulations,
we obtain easy-to-use criteria for the occurrence of the common envelope phase
in mass-transferring BH binaries with a nondegenerate donor, and incorporate
into population synthesis calculations. To explore the impact of possible mass
gap between NSs and BHs on the properties of merging BH--CS binary population,
we adopt different supernova mechanisms involving the \textit{rapid},
\textit{delayed} and \textit{stochastic} prescriptions to deal with the compact
remnant masses and the natal kicks. Our calculations show that there are $ \sim
10^{5} -10^{6}$ BH--CS binaries in the Milky Way, among which dozens are
observable by future space-based gravitational wave detectors. We estimate that
the local merger rate density of all BH--CS systems is $ \sim 60-200 \,\rm
Gpc^{-3}yr^{-1}$. While there are no low-mass BHs formed via \textit{rapid}
supernovae, both \textit{delayed} and \textit{stochastic} prescriptions predict
that $ \sim 100\% $/$ \sim 70\% $/$ \sim 30\% $ of merging BH--WD/BH--NS/BH--BH
binaries are likely to have BH components within the mass gap.
