Let $M$ be a non-compact manifold, equipped with a (closed?) submanifold $N\subset M$. The action of $Diff(M)$ on the set of embeddings $N\hookrightarrow M$ induces a map $$ Diff(M) \rightarrow Emb(N,M). $$ Is this map a fibration in the sense of Hurewicz?

I am aware of the results of Palais and lately Goodwillie in the case of compact manifolds, but I have no idea about the noncompact case.