My main source for currents and distribution theory on manifolds in general is de Rham's Differentiable Manifolds. To recap, let $M$ be a smooth, $m$ dimensional real manifold without boundary. De Rham deals with the nonorientable case by using pseudoforms but I don't want to go into that as well, so I assume $M$ to be orientable and oriented and further suppose that all manifolds or chains encountered in this question are also oriented.
Let $\mathfrak D_k=\mathfrak D_k(M)$ denote the space of smooth compactly supported $m-k$-forms on $M$ and $\mathfrak D_k^\ast=\mathfrak D_k^\ast(M)$ to be the space of all linear functionals on $\mathfrak D_k$ which are continuous in the sense that $\chi\in \mathfrak D_k^\ast$ if and only if for each sequence $\varphi_n\in\mathfrak D_k$ which are supported in a single compact set $K\subseteq M$ which is itself contained in a single coordinate domain such that the components of $\varphi_n$ and all partial derivatives of all orders converge uniformly to zero on $K$, then $\chi[\varphi_n]\rightarrow 0$, where $[\cdot]$ denotes the application of a functional to a field.
De Rham calls the elements of $\mathfrak D_k^\ast$ $k$-currents or currents of dimension $m-k$. A $k$-current $\omega$ is regular if there exists a $k$-form also denoted by $\omega$, not necessarily smooth or even continuous (but locally integrable) such that $\omega[\varphi]=\int_M\omega\wedge\varphi$ for all $\varphi\in \mathfrak D_k$. A current is singular otherwise.
In particular, de Rham defines the boundary $\partial\omega$ of a $k$-current $\omega$ to be the $k+1$-current given by $$ \partial\omega[\varphi]=\omega[d\varphi], $$and defines the exterior derivative $d\omega$ of a $k$-current by $d\omega=(-1)^{k+1}\partial\omega$. The exterior derivative then agrees with the usual exterior derivative if $\omega$ is regular and is represented by a $C^1$ form.
It seems that $M$ having no boundary is crucial to the theory, in fact the ordinary theory of Schwartz distributions on $\mathbb R^m$ also relies crucially on the fact that when test functions are involved, one can integrate by parts with reckless abandon.
For example it is not difficult to see that if $M$ has a boundary and one wants to keep consistency with the regular case, the relationship between boundaries and exterior derivatives need to be modified. For example let $\omega\in\mathfrak D_k^\ast(M)$ be a $k$-current represented by a $C^\infty$ form (also denoted $\omega$), then on any test $m-k-1$-form $\varphi$ we have $$ \partial\omega[\varphi]=\omega[d\varphi]=\int_M\omega\wedge d\varphi=(-1)^{k+1}\int_Md\omega\wedge\varphi+(-1)^k\int_Md(\omega\wedge\varphi) \\ =(-1)^{k+1}\int_Md\omega\wedge\varphi+(-1)^k\int_{\partial M}\omega\wedge\varphi, $$ thus we can write $$ \partial\omega=(-1)^{k+1}d\omega+(-1)^k\delta_{\partial M}\wedge\omega, $$ where $\delta_{\partial M}$ is the "Dirac delta $1$-current" defined by $\delta_{\partial M}[\varphi]=\int_{\partial M}\varphi$.
As far as I can tell the Dirac delta current is well-defined (it is continuous and the embedding of $\partial M$ in $M$ is proper, so the pullback of $\varphi$ is also compactly supported), but the product $\delta_{\partial M}\wedge\omega$ is only well-defined if $\omega$ is $C^\infty$ (or at least $C^0$ - I know the ordinary point-Dirac delta is well-defined on continuous test functions as well and I think the same should hold for this $\delta_{\partial M}$ but I am not sure right now).
It thus seems to me that if $M$ has boundary then the exterior derivative of a current is ill-defined and even if it does make sense, it does not coincide with the boundary (up to sign that is).
I suspect this is not the only difference between the cases when $M$ has no boundary and when $M$ has boundary.
I am therefore looking for papers or textbooks which consider either de Rham currents or some other formulation of distribution theory on manifold with boundaries or corners.
