The following construction is standard, and it deserves a name.

Suppose we need to construct a diffeomorphism from a manifold $M$ to itself with some additional properties. Observe that the flow $\phi^t$ for any reasonable vector field $v(t)$ on $M$ defines a diffeomorphism for any $t$. So it remains to find a vector field $v(t)$ such that the flow $\phi^1$ satisfies your properties.

Do you know a name for it?

If not, what would be a good name?