  Motivated by recent $R_D$, $R_{D^*}$ and $R_{J/\psi}$ anomalies in $B\to
D\ell^- \bar{\nu}_{\ell}$, $B\to D^*\ell^- \bar{\nu}_{\ell}$ and $B_c\to
J/\psi\ell^- \bar{\nu}_{\ell}$ decays, respectively, we study possible R-parity
violating supersymmetric effects in $B_c\to J/\psi\ell^-
\bar{\nu}_{\ell},\eta_c\ell^- \bar{\nu}_{\ell}$ and $\Lambda_b \to \Lambda_c
\ell^- \bar{\nu}_{\ell}$ decays, which are also induced by $b\to c \ell^-
\bar{\nu}_{\ell}$ at quark level. We find that (I) the constrained slepton
exchange couplings $\lambda_{i33}\tilde{\lambda}'^*_{i23}$ involving in $b\to c
\tau^- \bar{\nu}_{\tau}$ transition from relevant latest experimental data
still have quite large effects on all (differential) branching ratios and the
normalized forward-backward asymmetries of the exclusive semileptonic $b\to c
\tau^- \bar{\nu}_{\tau}$ decays as well as the ratios of the (differential)
branching ratios; (II) after satisfying the data of $R_D$ and $R_{D^*}$, the
upper limit of $R_{J/\psi}$, $R_{\eta_c}$ and $R_{\Lambda_c}$ could be
increased by 10%, 112% and 24%, respectively, from their upper limits of the
Standard Model predictions by the $\lambda_{i33}\tilde{\lambda}'^*_{i23}$
couplings. The results in this work could be used to probe R-parity violating
effects and will correlate with searches for direct supersymmetric signals at
the running LHCb and the forthcoming Belle-II.
