Name for a standard trick to construct a diffeomorphism 
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?

 A: This is often called "path method". In particular one has classically the Moser path method.
Let us prove for example (after Moser) that given a closed oriented $n$-manifold $M$ and two volume forms $\omega_0$, $\omega_1$ on $M$ with the same integral, there exists a diffeomorphism $f$ of $M$ such that $$f^*(\omega_1)=\omega_0$$ To this end, the path method considers $$\omega_t=(1-t)\omega_0+t\omega_1$$ and looks for  a time-dependant vector field $X_t$ whose flow
$\phi_t$ satisfies the condition $$\phi_t^*(\omega_t)=\omega_0$$ (Then, for $t=1$, $f=\phi^1$ will work)
The condition holds trivially for $t=0$. Deriving this condition with respect to $t$, one finds $$\phi_t^*(L_{X_t}\omega_t+\omega_1-\omega_0)=0$$ where $L$ is the Lie derivative. This amounts to $$L_{X_t}\omega_t=\omega_0-\omega_1$$ i.e. by Cartan's formula
$$d\iota_{X_t}\omega_t=\omega_0-\omega_1$$
Since $\omega_0$ and $\omega_1$ have the same integral, they are cohomologous: one has a $(n-1)$-form $\alpha$ on $M$ such that $$\omega_0-\omega_1=d\alpha$$ Hence it is enough that 
$$\iota_{X_t}\omega_t=\alpha$$
The end of the argument is purely (multi)linear algebra: this last equation admits for every time $t$ and at every point $x$ a unique solution $X_t(x)$ since $\omega_t(x)$ is a nonzero $n$-form on $T_xM$.
An analogous method applies to symplectic forms, nonsingular closed $1$-forms and contact forms.
The path method also proves the $MJ^2$ Lemma (Lemma 3.2 in F. Laudenbach, A proof of Reidemeister-Singer's theorem by Cerf's methods. Ann. Fac. Sci. Toulouse Math. (6) 23 (2014), 1, 197–221, Arxiv 1202.1130); and in particular the Morse Lemma.
A: This is a hugely studied field in the area of medical image processing. Goes by name of (large) diffeomorphic registration or diffeomorphic mapping
https://en.m.wikipedia.org/wiki/Large_deformation_diffeomorphic_metric_mapping
