Let X,Y be smooth manifolds and let $f:X \to C^{-\infty}(Y)$ be a continuous map, where $ C^{-\infty}(Y)$ is the space of generalized functions on $Y$ equipped with the weak topology. By Schwartz kernel theorem, this gives us a generalized function $\xi_f$ on $X \times Y$. Do you know any reference for this construction and its properties?

Specifically, what is the relationship between this construction and the wave front set? For example, if we know the wave front set of $\xi_f$, it seems that we can bound the wave front set of $f(x)$. Is it written anywhere?

In general, it is reasonable to say that the restriction of $\xi_f$ to $\{x\}\times Y$ is $f(y)$. On the other hand there another is a notion of restriction (or more generally pull back) of a distribution (see Hormander The analysis of Linear PD operators I, Theorem 8.2.4 ). Do you know any reference for the fact that those to notions coincide whenever both are defined?