What we can say is
$\begin{array}{c l} \nabla_{\rm x}\cdot\int_D \mathrm{F(x,r)}dV & =\nabla_{\rm x}\cdot\int_D\big(f_1(\mathrm{x,r}),\cdots,f_n(\mathrm{x,r})\big) dV \\ & = \nabla_{\rm x}\cdot\left(\int_Df_1(\mathrm{x,r})dV,\cdots,\int_Df_n(\mathrm{x,r})dV\right) \\ & = \frac{\partial}{\partial x_1}\left(\int_Df_1(\mathrm{x,r})dV\right)+\cdots+\frac{\partial}{\partial x_n}\left(\int_Df_n(\mathrm{x,r})dV\right) \\ & = \int_D \frac{\partial}{\partial x_1}f_1(\mathrm{x,r})dV+\cdots+\int_D\frac{\partial}{\partial x_n}f_n(\mathrm{x,r})dV \\ & = \int_D\left( \frac{\partial}{\partial x_1}f_1(\mathrm{x,r})+\cdots+\frac{\partial}{\partial x_n}f_n(\mathrm{x,r})\right)dV \\ & = \int_D\nabla_{\rm x}\cdot\mathrm{F(x,r)}dV. \end{array}$
Remarks.
- $\mathrm{x}=(x_1,\cdots,x_n)$ is a vector and its components.
- $\mathrm{F(x,r)}=\big(f_1(\mathrm{x,r}),\cdots,f_n(\mathrm{x,r})\big)$ is a vector function of both $\rm x$ and $\rm r$. It is $\Bbb R^n\times D\to\Bbb R^n$.
- $\nabla_{\rm x}\,\cdot=\mathrm{div}_{\rm x}$ is with respect to $\rm x$ and only applies to vector functions, not scalar functions.
- The integration over the region $D$ involves the dummy vector $\rm r$.
- This works whenever the partial differentials and integration can be interchanged.
Also, similarly,
$\nabla_{\rm x} \int_D f(\mathrm{x,r})dV=\int_D \nabla_{\rm x} f(\mathrm{x,r})dV$
for a scalar function $f$. Note the difference between the divergence $\nabla\cdot$ and gradient $\nabla$.