Suppose $S\subset K$ is an embedded oriented $k$-submanifold of orinetable smooth manifold $M$. Let $\Phi$ be a flow of some complete smooth vector field $X$ on $M$. Then, $$\int_{S}\Phi_t^*\eta =\int_{\Phi_t(S)}\eta,$$ for a smooth $k$-form with compact support on $M$ and $${d\over dt}\int_{\Phi_t(S)}\eta = \int_{\Phi_t(S)}X\lrcorner d\eta+d(X\lrcorner\eta).$$
For the first equality, let $\{\psi_i\}$ be a smooth partition of unity subordinate to positively oriented smooth charts $\{(U_i,\varphi_i)\}$ of $M$. Then, \begin{align*} \int_S\Phi_t^*\eta & = \sum_i\int_S\psi_i\Phi_t^*\eta\\ & = \sum_i\int_{\hat{U}_i}(\varphi_i^{-1})^*(\psi_i\Phi_t^*\eta)\\ & = \sum_i\int_{\hat{U}_i}\psi_i(\varphi_i^{-1})^*(\Phi_t^*\eta)\\ & = \sum_i\int_{\hat{U}_i}\psi_i(\Phi_t\circ\varphi_i^{-1})^*\eta\\ & = \sum_i\int_{\Phi_t(S)}\psi_i\eta\\ & = \int_{\Phi_t(S)}\eta\\ \end{align*} Here, $\Phi_t$ is positively oriented as $\Phi_0 = 1$ and by continuity on $t$, if $\omega$ is a positively oriented orientation form on $M$ then $\Phi_t^*\omega$ is a positively oriented orientation form on $M$.
For the second equality, of course, we use Cartan's formula. \begin{align*} {d\over dt}\bigg|_{t =t_0}\int_{\Phi_t(S)}\eta & = {d\over dt}\bigg|_{t =t_0}\int_S\Phi_t^*\eta\\ & {\color{red} =}\int_S{d\over dt}\bigg|_{t =t_0}\Phi_t^*\eta\\ & = \int_S\Phi_{t_0}^*(\mathcal{L}_X\eta)\\ & = \int_{\Phi_{t_0}(S)}\mathcal{L}_X\eta\\ & = \int_{\Phi_{t_0}(S)}X\lrcorner d\eta+\int_{\Phi_{t_0}(S)}d(X\lrcorner\eta)\\ \end{align*}
I'm not sure the red-colored equality is allowed. Can anyone please confirm I did it right?