The two excited vector charmonium states $Y\left(4220\right)$ and $Y\left(4360\right)$ are difficult to understand as pure $c\overline{c}$ charmonium states. Since they are located close to the mass thresholds of $\overline{D}{D}_1$ and ${\overline{D}}^{*}{D}_1$, they can be viewed as $\overline{D}{D}_1$ and ${\overline{D}}^{*}{D}_1$ molecules. Furthermore, recent studies indicate that the exotic states $X\left(3872\right)/Z_c\left(3900\right)$ and $X_2\left(4013\right)/Z_c\left(4020\right)$ are the isoscalar/isovector $\overline{D}{D}^{*}$ and isoscalar/isovector ${\overline{D}}^{*}{D}^{*}$ molecules, respectively. In this work, in the molecular picture, we employ the triangle diagram mechanism to study the productions of $Z_c\left(3900\right)$ and $X\left(3872\right)$ in the pionic and radiative decays of $Y\left(4220\right)$, as well as their heavy-quark spin symmetry partners—i.e., the productions of $Z_c\left(4020\right)$ and $X_2\left(4013\right)$ in the pionic and radiative decays of $Y\left(4360\right)$. Using the effective Lagrangian approach, we obtain the ratios of the branching fractions $B\left[Y\right(4360)\to Z_c(4020\left)\pi \right]/B\left[Y\right(4220)\to Z_c(3900\left)\pi \right]=1.2\pm 0.3$ and $B\left[Y\right(4360)\to X_2(4013\left)\gamma \right]/B\left[Y\right(4220)\to X(3872\left)\gamma \right]=0.5\pm 0.1$, almost independent of model parameters, which indicate that the productions of $X_2\left(4013\right)$ and $Z_c\left(4020\right)$ in the radiative and pionic decays of $Y\left(4360\right)$ are likely to be measured in the future. The experimental studies of the predicted decay modes will help to verify the molecular nature of $X\left(3872\right)$, $Z_c\left(3900\right)$, and $Y\left(4220\right)$. We hope that the present work can stimulate experimental and further theoretical studies on these decay modes.