  We study the weak decays of $\bar{B}_{(s)}$ and $B_c$ into $D$-wave
heavy-light mesons, including $J^P=2^-$ ($D_{(s)2},D'_{(s)2},B_{(s)2},
B'_{(s)2}$) and $3^-$~($D^*_{(s)3}, B^*_{(s)3}$) states. The weak decay
hadronic matrix elements are achieved based on the instantaneous Bethe-Salpeter
method. The branching ratios for $\bar{B}$ decays are $\mathcal{B}[\bar{B}\to
D_{2}e\bar{\nu}_e] = 1.1^{-0.3}_{+0.3} \times 10^{-3}$, $\mathcal{B}[\bar{B}\to
D'_2e\bar{\nu}_e]=4.1^{-0.8}_{+0.9} \times 10^{-4}$, and
$\mathcal{B}[\bar{B}\to D^*_3e\bar{\nu}_e]=1.0^{-0.2}_{+0.2} \times 10^{-3}$,
respectively. For semi-electronic decays of $\bar B_s$ to $D_{s2}$, $D'_{s2}$,
and $D^*_{s3}$, the corresponding branching ratios are $1.7^{-0.5}_{+0.5}\times
10^{-3}$, $5.2^{-1.5}_{+1.6}\times 10^{-4}$, and $1.5^{-0.4}_{+0.4}\times
10^{-3}$, respectively. The branching ratios of semi-electronic decays of $B_c$
to $D$-wave $D$ mesons are in the order of $10^{-5}$. We also achieved the
forward-backward asymmetry, angular spectra, and lepton momentum spectra. In
particular the distribution of decay widths for $2^-$ states $D_2$ and $D'_2$
varying along with mixing angle are presented.
