Let $M$ be a smooth manifold and let $f_0, f_1 \colon [0, 1] \times \partial M \to M$ be two smooth embeddings that are the identity map on $\partial M \times\{0\} = \partial M$ . If $\partial M$ is

non-compactdoes there exist a diffeotopy $H \colon M × [0, 1] \to M$ relative to $\partial M$ from $H_0=\text{id}_M$ to a diffeomorphism $H_1\colon M\to M$ with $H_1\circ f_0=f_1$?

*The case for compact boundary can be found in most textbooks.* Can someone suggest to me some references for the non-compact boundary case?