Let $M$ be a smooth oriented manifold of dimension $n$ and $T$ a current of dimension $k$ on $M$. Let $\phi:P\times M \to M$ be a proper smooth family of diffeomorphisms of $M$ (i.e. $P$ is a smooth manifold, $\phi$ is a smooth proper map, and each $\phi_p:=\phi(p, \cdot)$ is a diffeo of $M$) and $dp$ a smooth measure on $P$. Suppose that $\phi(\cdot,x)$ is a submersion for each $x \in M$.
I claim that the current $S:= \int_P \phi_{p*} T \, dp$ (i.e. $S(\alpha) = \int_P T(\phi_p^*\alpha) \, dp$ for $\alpha$ a $k$-form) is smooth, i.e. is given by integration against a smooth $(n-k)$-form. This should be a standard fact. I think I can give a proof but it is tedious. Does anyone know a reference?
I would be content with a reference covering the case where $T$ is given by integration over a smooth oriented submanifold, but surely this is an artificial restriction.