I am looking for some geometrical insights into the Reynolds transport theorem. Firstly, the case when looking at the transport of scalar quantities is rather trivial, let $\rho \omega$ be a spatial k-form on $M^k$, the integral over time-dependent region $\Omega(t) = \psi_{t,t_0} (\Omega(t_0)) \subset M$ then reads
\begin{equation} \int_{\Omega(t)} \rho \omega = \int_{\Omega(t)} \frac{\partial \rho} { \partial t} \omega + L_{\mathbf{u}} (\rho \omega) = \int_{\Omega(t)} \frac{\partial \rho} { \partial t} \omega + \int_{\partial \Omega(t)} \iota_{\mathbf{u}}(\rho \omega) \end{equation}
where $\mathbf{u}=\dot{\psi}$ and Cartans magic formula was used. However, when considering the transport of vector-valued quantities (such as momentum) I am a bit stuck. I don't really know how to interpret/ derive the following
\begin{equation} \int_{\Omega(t)} \mathbf{f} \omega = \int_{\Omega(t)} \frac{\partial \mathbf{f}} { \partial t} \omega + \int_{\partial \Omega(t)} (\mathbf{f} \otimes \mathbf{u}) \cdot \mathbf{n} \;\iota_{\mathbf{n}}\omega. \end{equation}
Intuitively I would assume that we must require $M$ to be flat, such that we can map all the distinct vector spaces at $TM_x$ to some reference point through parallel transport, where we may then sum over them. Still, I am very puzzled how to pull back the domain $\Omega(t)$ to $\Omega(t_0)$. Taking inspiration from the scalar case I would assume that we retrieve the Lie-derivative $L_{\mathbf{u}} \mathbf{f}$, however, we do not. Furthermore, how can we show that the resulting transport theorem is coordinate-independent?
I hope the answer is not too obvious. I would be grateful for any references or help.