Equation of Motion from Vlasov Equation From the Vlasov equation
$$\frac{\partial f}{\partial t} + \vec{v} \cdot \nabla f + \vec{a} \cdot \frac{\partial f}{\partial \vec{p}} = 0$$
we can obtain the momentum transport equation by multiplying by $\vec{p}$ and integrating over the space of momenta,
$$\int \mathrm{d}^3p \left[\vec{p}\frac{\partial f}{\partial t} + \vec{p}\vec{v} \cdot \nabla f + \vec{p}\vec{a} \cdot \frac{\partial f}{\partial \vec{p}}\right] = 0$$
This derivation is readily available in the literature, and every book or source I have checked so far calculates the first term as
$$\int \mathrm{d}^3p \left(\vec{p}\frac{\partial f}{\partial t}\right) = \int d^3p \left(\frac{\partial \vec{p}f}{\partial t}\right) = m\frac{\partial }{\partial t}\int d^3p \left(\vec{v}f\right) = m\frac{\partial }{\partial t}(n\vec{u})$$
I don't understand why it is possible to simply put $\vec{p}$ inside the derivative; isn't it a function of $t$? 
I have thought about considering the functions as defined on a space $(\vec{q},\vec{p},t)$ which would make $\vec{p}$ and $t$ independent variables (and thus $\frac{\partial\vec{p}}{\partial t} = 0$), but I do not actually have a background in either Lagrangian mechanics or statistical physics, so I am unsure if that even makes any sense. For one, it doesn't seem right at first glance that the momentum would be independent of time like that.
Is there a simplification being done that I am simply not seeing, or is there a deeper reason behind this result being valid?
EDIT: The relation
$$\vec{u} = \frac{1}{n} \int \mathrm{d^3 p} ~ \vec{v} f$$
and where it comes from is clear to me, the problem I'm having is only in the manipulation of the time derivative inside the integral.
 A: 
I don't understand why it is possible to simply put $\vec{p}$ inside the derivative; isn't it a function of t?

No, $\vec{p}$ is not a function of $t$.  The $\vec{p}$ are momentum coorinates/abscissa of $f$, referred to as a random variable, not free variables in the system, as it were.  Thus, the commutation of the partial derivative operation is totally fine.
As an example, try thinking about this from a discretized point of view.  Imagine you have some probability function, $g = g(x,t)$, that is known on some discrete grid at any given time, $t$.  The grid itself does not change in time, it's just the locations in physical space where $g$ is evaluated.  The values of $g$, however, are not independent of time and can change.
In the limit as the number of grid points goes to infinity, the time independence of $x$ does not change.  The "grid" just becomes a continuous construct with $dx$ being infinitesimal in magnitude.

Is there a simplification being done that I am simply not seeing, or is there a deeper reason behind this result being valid?

It's not a deep, metaphysical issue or anything, just a consequence of the construction of the problem and assumptions that go with it.
