= Solution
Put $z_i=(\mathbf x_i,\mathbf v_i)$ and $dz_i=d^3x_i\,d^3v_i$. Indistinguishability gives an <exchangeable> ensemble. Define unit-normalized <marginal distributions>
$$
p_1(z_1,t)=\int\varrho\,dz_2\cdots dz_N,\qquad
p_2(z_1,z_2,t)=\int\varrho\,dz_3\cdots dz_N.
$$
Integrating the <dissipative Liouville equation> over $z_2,\ldots,z_N$ removes their boundary fluxes. The $N-1$ identical pair terms remain:
$$
\partial_tp_1+\mathbf v\cdot\nabla_{\mathbf x}p_1
+(N-1)\nabla_{\mathbf v}\cdot\int K(\mathbf x,\mathbf x_\star)(\mathbf v_\star-\mathbf v)p_2(z,z_\star,t)\,dz_\star=0.
$$
This is the first equation of a <BBGKY hierarchy>, with no independence assumed yet.
The additional physical assumption is a <mean-field closure for velocity alignment>: neglect pair correlations, $p_2(z,z_\star,t)\simeq p_1(z,t)p_1(z_\star,t)$. Initial independence alone does not make this exact at later times, because the particles interact. In the usual number convention, $f=Np_1$ and the <two-particle distribution function> is $f_2=N(N-1)p_2$. Consequently the closed equation at finite $N$ is
$$
\partial_tf+\mathbf v\cdot\nabla_{\mathbf x}f
+\left(1-\frac1N\right)\nabla_{\mathbf v}\cdot\bigl(f\mathbf L[f]\bigr)\simeq0,
\qquad
\mathbf L[f](z)=\int K(\mathbf x,\mathbf x_\star)(\mathbf v_\star-\mathbf v)f(z_\star,t)\,dz_\star.
$$
Neglecting the relative $1/N$ correction gives the intended <mean-field alignment equation>:
$$
\boxed{\partial_tf+\mathbf v\cdot\nabla_{\mathbf x}f+\nabla_{\mathbf v}\cdot\bigl(f\mathbf L[f]\bigr)=0.}
$$
Equivalently, the closure $f_2\simeq ff_\star$ incorporates this large-$N$ approximation. It cannot be an exact finite-$N$ identity with $\int f=N$, since $\int f_2\,dz_\star=(N-1)f$ whereas $\int ff_\star\,dz_\star=Nf$.
There is also a consistent finite-$N$ convention yielding the boxed equation directly under the independence closure: define $\widetilde f=(N-1)p_1$ instead. Its total <normalization of a probability distribution> is $N-1$, rather than the physical particle number $N$. Thus the displayed source equation needs a stated <normalization of a probability distribution> convention, or the large-$N$ approximation, in addition to neglect of correlations. In the remaining parts we use the displayed <mean-field alignment equation> itself; its moment identities do not depend on the total <normalization of a probability distribution>.
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