A compactly supported time-dependent field has a global time-one flow

The factor swap makes $V(t,y)=G(y,t)$ smooth, with $V_t=G_t$, so [F1] applies to $V$. If $N$ is boundaryless, [L1] gives a global smooth evolution on $I$. For the boundary case, in a boundary chart write the inward coefficient as $a(y^{\mathrm{\prime }},r,t)$, where $r\ge 0$. Tangency gives $a(y^{\mathrm{\prime }},0,t)=0$, and local smooth extension gives $a=rb$ with $b(y^{\mathrm{\prime }},r,t)={\int }_0^1{\mathrm{\partial }}_{r}a(y^{\mathrm{\prime }},ur,t)du$ smooth. Extend the coordinate field across $r=0$ and apply the local smooth ODE theory underlying [L1]. Uniqueness keeps solutions starting on $r=0$ there; for $r(s)>0$, the scalar equation gives $r(t)=r(s)\exp ({\int }_s^{t}b(y^{\mathrm{\prime }}(u),r(u),u)du)>0$ while the solution is in the chart. Thus local solutions and their reverse-time solutions preserve the half-space. Global continuation is the compact-support argument of [L1]: a solution meeting the complement of the common compact support set is constant by uniqueness; any other solution stays in that compact set. At a finite maximal endpoint a sequence of its values has a convergent subsequence there, and a local evolution around the limiting time and point extends the solution by uniqueness. This works also at a boundary point using the half-space solutions just established, and at $t=0,1$ using local smooth time extension. Hence solutions exist on all of $I$ with smooth dependence. Their initial-value identity and [L2] give properties 1 and 3.

Property 2: composing the cocycle law with $r=s$ gives ${\mathrm{\Psi }}_{s,t}\circ {\mathrm{\Psi }}_{t,s}={\mathrm{\Psi }}_{s,s}={\mathrm{i}\mathrm{d}}_N$, and with the roles of $s,t$ exchanged gives ${\mathrm{\Psi }}_{t,s}\circ {\mathrm{\Psi }}_{s,t}={\mathrm{i}\mathrm{d}}_N$; hence each ${\mathrm{\Psi }}_{t,s}$ is a bijection with inverse ${\mathrm{\Psi }}_{s,t}$, and both are smooth by [L1], so each ${\mathrm{\Psi }}_{t,s}$ is a diffeomorphism of $N$ (Diffeomorphisms and local diffeomorphisms of manifolds).

Property 4 and uniqueness: suppose $G$ vanishes identically on $[s,t]$. The constant curve $r\mapsto y$ solves the initial-value problem with value $y$ at time $s$, and so does $r\mapsto {\mathrm{\Psi }}_{r,s}(y)$; by uniqueness of integral curves [L3] the two agree, whence ${\mathrm{\Psi }}_{t,s}(y)=y$ for every $y$, i.e. ${\mathrm{\Psi }}_{t,s}={\mathrm{i}\mathrm{d}}_N$. Uniqueness of the evolution operator itself is the same statement: any evolution operator satisfying the initial-value problem has the same integral curves as the one constructed in step 1.1, so it agrees with it everywhere.

Setting $H_t:={\mathrm{\Psi }}_{t,0}$ gives a smooth family of diffeomorphisms with $H_0={\mathrm{i}\mathrm{d}}_N$ and $H_t^{-1}={\mathrm{\Psi }}_{0,t}$ by step 2.1; since ${\mathrm{\Psi }}_{t,0}$ is the identity outside the compact set containing ${\bigcup }_t\mathrm{supp}{G}_t$ (a point outside the supports has the constant curve as its integral curve, by [L3] as in step 2.2), $H$ is a compactly supported ambient isotopy, and it is stationary on every interval on which $G$ vanishes by step 2.2.