Let $M$ and $W$ be smooth manifolds (possibly with boundary) and $V\subseteq W$ a submanifold. We have a map between embedding spaces
$$Emb(W,M)\rightarrow Emb(V,M)$$ given by restriction.

[Richard Palais proved around 1960][1] that if all manifolds have no boundary, $V$ is compact and the embedding spaces are equipped with the weak Whitney $C^{\infty}$-topology, then this is a fiber bundle, in particular a Serre fibration. (The path-lifting property of the fibration reflects the isotopy extension theorem.)

In conclusive remarks (section 6 of the cited paper), he mentions that the result is also true allowing $W$ and $V$ to have boundary but still insisting on $V$ being compact and $M$ having no boundary. It is said that these results will appear, among others, in a joint paper with Morris Hirsch.

It seems to me that this paper was never published.

 1. Has a proof of the announced result been published elsewhere in the meanwhile?
 2. Does it fail for $M$ having boundary or $V$ being noncompact? If so, is the map still a fibration? I am especially interested in the case of $V,M,W$ all being compact and with boundary.

  [1]: http://vmm.math.uci.edu/PalaisPapers/LocalTrivOfRestrictionForEmbeddings.pdf