Let $\vartheta_X$ denote a positive smooth function on a compact Kähler manifold $M$. The subscript $X$ denotes a holomorphic vector field and we have the following relation $$i_X \omega = - \overline{\partial} \vartheta_X,$$ the function $\vartheta_X$ comes from the Hodge decomposition. The harmonic part has already been shown to be zero and $\overline{\partial} (i_X \omega )=0$.
Let $\omega$ be a fixed background Kähler metric on $M$and let $\omega_t = \omega + \partial \overline{\partial} \varphi_t$, where $\varphi_t$ is a smooth $\mathbb{R}$-valued function.
Let $\Delta_t$ denote the Laplacian with respect to the metric $g_t$, i.e., $$\Delta_t = \sum_{i,j=1}^n g_t^{i \overline{j}} \frac{\partial^2}{\partial z^i \partial \overline{z}^j},$$ where $g_t : = (g + \partial \overline{\partial} \varphi_t)$. Moreover, let $\vartheta_{X, t} = \vartheta_{X,0} - X(\varphi_t)$.
I would like to verify a computation of $ \dfrac{\partial}{\partial t} \Delta_t \vartheta_{X,t}$. To this end,
\begin{eqnarray*} \frac{\partial}{\partial t} \Delta_t \vartheta_{X,t} &=& \sum_{i,j=1}^n \frac{\partial}{\partial t} \left( g_t^{i \overline{j}} \frac{\partial^2}{\partial z^i \partial \overline{z}^j} (\vartheta_{X,0} -X(\varphi_t)) \right) \\ &=& \sum_{i,j=1}^n \frac{\partial}{\partial t} \left( g_t^{i \overline{j}} \frac{\partial^2 \vartheta_{X,0}}{\partial z^i \partial \overline{z}^j} - g_t^{i \overline{j}} \frac{\partial^2}{\partial z^i \partial \overline{z}^j} X(\varphi_t) \right) \\ &=& - \sum_{i,j=1}^n \frac{\partial}{\partial t} \left( g_t^{i\overline{j}} \frac{\partial^2}{\partial z^i \partial \overline{z}^j} X(\varphi_t) \right) \\ &=& - \sum_{i,j=1}^n \left( - \partial \overline{\partial} \dot{\varphi_t} \left( -g_t^{j \overline{k}} g_t^{i \overline{j}} \frac{\partial (g_t)_{j \overline{k}}}{\partial t} \right) \frac{\partial^2}{\partial z^i \partial \overline{z}^j} X(\varphi_t) \right) \\ && - \sum_{i,j=1}^n g_t^{i \overline{j}} \dot{\varphi_t} \frac{\partial^2}{\partial z^i \partial \overline{z}^j} \dot{X}(\varphi_t) \\ &=& - (\partial \overline{\partial} \dot{\varphi}_t) (\nabla_t g_t) \Delta_t \vartheta_{X,t} - \dot{\varphi}_t \Delta_t \dot{X}(\varphi_t) \end{eqnarray*}