I am looking to replicate, in the fiberwise setting, the result of Spivak/Wall that the fiber homotopy type of the normal sphere bundle of a manifold is preserved under homotopy equivalences.

A particularly slick way to prove this result, is to notice that the mapping cylinder of a homotopy equivalence $M \rightarrow M'$ is a relative Poincare duality space, and as such we can talk about its normal fibration (obtained from a relative embedding into $\mathbb{R}^N \times (0,1]$), which restricts to the normal fibrations of $M$ and $M'$ on either side. Thus, because a mapping cylinder of a homotopy equivalence has both end inclusions homotopy equivalences, we can conclude there is some fiberwise homotopy equivalence of normal spherical fibrations.

So here is my question: is there is enough fiberwise embedding theory in the literature to be able to conclude that the fiberwise mapping cylinder of a fiber homotopy equivalence of manifold bundles over $B$ embeds into $B \times (\mathbb{R}^{N} \times (0,1])$, such that $M,M'$ land in the boundary?