A fibred manifold is a triple $(E,\pi,M)$ where $E$ and $M$ are manifolds and $\pi : E \rightarrow M$ is a surjective submersion. (Saunders, The Geometry of Jet bundles)

A special case of this is the fibre bundle which is a fibred manifold, where for each relatively open subset $U \subset M$ the space $\pi^{-1}(M)$ looks like (i.e. is diffeomorphic to) a product $U \times F$ for some smooth manifold $F$.

Even more special are linear bundles and, of course, the tangential bundle of a smooth manifold.

I do not know a generalization of these constructions to manifolds with boundary. For example, it seems intuitive that at a boundary point of a smooth manifold with boundary, we do not have a tangential space, but rather a tangential cone. Similarly one would adapt the more general classes of fibre bundles.

Unfortunately, this seems to be non-standard in differential geometry and differential topology, and while it seems natural to me to work with cones in that case, reinventing the wheel might be a waste of time. Do you know a good reference that settles the basic constructions and possible pitfalls for "fibred manifolds with boundary"?