$$\newcommand{\div}{\operatorname{div}}$$ Let $M$ be a smooth compact, connected manifold with non-empty boundary.
Let $X$ be a smooth vector field on $M$ and suppose that the flow $\phi_t$ associated with $X$, is defined on all $M$, for sufficiently small $t>0$.
Is it true that $\phi_t:M \to M$ a smooth embedding?
I think that this is the case if $X$ is either tangent to $\partial M$ at every point of $\partial M$, or $X$ is strictly inward-pointing everywhere on $\partial M$;
I am not sure what happens when $X$ is tangent at some points of $\partial M$, and inward-pointing on other points.