A smooth isotopy of a compact manifold extends to an ambient isotopy

Compactness of the image and a relatively compact neighbourhood. If $M=\mathrm{\varnothing }$, take $H_t={\mathrm{i}\mathrm{d}}_N$ for every $t$; all extension and support assertions are then immediate. Assume $M\mathrm{\ne }\mathrm{\varnothing }$. The set $K:=F(M\times I)\subseteq N$ is compact, being the continuous image of the compact space $M\times I$, and it is closed in $N$ because smooth manifolds are Hausdorff. Each point of $K$ has a coordinate ball whose closure is compact and contained in $W$; finitely many of these balls cover $K$, and their union $V$ is an open neighbourhood of $K$ with compact closure $\bar{V}\subseteq W$. Fix such a $V$.

The velocity field along the slices, as a field along a graph. Extend $F$ to $\mathbb{R}\times M$ by $F_t=F_0$ for $t<0$ and $F_t=F_1$ for $t>1$. This extension is smooth because the given $F$ is constant on the full endpoint collars of width $\epsilon$, and every extended slice remains an embedding. Put $J:=\mathbb{R}$ and consider the map $\mathrm{\Theta }\text{\hspace{0.17em}⁣}:J\times M\to J\times N$, $\mathrm{\Theta }(t,x):=(t,F_t(x))$, with image $S:=\mathrm{\Theta }(J\times M)$. $\mathrm{\Theta }$ is injective because its first coordinate is $t$; its derivative at $(t,x)$ equals $(\mathrm{d}t,{\mathrm{\partial }}_{t}F(x,t)\mathrm{d}t+\mathrm{d}(F_t{)}_x)$, which is injective because $\mathrm{d}(F_t{)}_x$ is injective by [F6]; and $\mathrm{\Theta }$ is proper: for a compact subset $L\subseteq J\times N$, its time projection is compact, and ${\mathrm{\Theta }}^{-1}(L)$ is closed in the compact product of that projection with $M$. The inverse on the image is continuous locally by the embedding property of its slices, or globally by this properness. Its injective derivative therefore makes $S$ a closed embedded submanifold without boundary of $J\times N$, of dimension $1+\dim M$. The constant time extension avoids applying a boundaryless extension theorem to a graph with boundary. The assignment $Y(t,F_t(x)):=(0,{\mathrm{\partial }}_{t}F(x,t))\in T_{(t,F_t(x))}(J\times N)$ is well defined because each $F_t$ is injective, and it is a smooth vector field along $S$: near a point of $S$ the inverse $y\mapsto x$ of $F_t$ is smooth by [F6], so $Y$ is the composite of smooth maps; along $S$ it is everywhere tangent to the splitting of $T(J\times N)$ into the $J$-direction and $TN$.

Extension and truncation. By [F4] applied to the closed embedded submanifold $S$ of the smooth manifold $J\times N$, the field $Y$ extends to a global smooth vector field $\tilde{Y}$ on $J\times N$ with $\tilde{Y}{\mathrm{\mid }}_S=Y$. Write $\tilde{Y}=(a,X)$ in the splitting $T(J\times N)\cong \mathbb{R}\oplus TN$, so that $X$ is a smooth family $X_t:=X(t,\cdot )$ of vector fields on $N$ with $X_t(F_t(x))={\mathrm{\partial }}_{t}F(x,t)$ for all $x\in M$ and $t\in J$. Choose a smooth function $\psi \text{\hspace{0.17em}⁣}:N\to [0,1]$ with $\psi =1$ on a neighbourhood of $K$ and $\mathrm{supp}\psi \subseteq V$: the open sets $V$ and $N\setminus K$ cover $N$ because $K$ is closed, so [F2] applied to this two-element cover produces $\psi$ as the member subordinate to $V$, whose support lies in $V$. The other member has closed support contained in $N\setminus K$; its support complement is an open neighbourhood of $K$ on which that other member vanishes and hence $\psi =1$. Choose a smooth $\beta \text{\hspace{0.17em}⁣}:\mathbb{R}\to [0,1]$ with $\beta =1$ on $[\epsilon ,1-\epsilon ]$ and $\beta =0$ on $(-\mathrm{\infty },\frac{\epsilon }{2}]\cup [1-\frac{\epsilon }{2},\mathrm{\infty })$, and put $X_t^{\mathrm{\prime }}:=\beta (t)\psi X_t$ for $t\in J$. Every $X_t^{\mathrm{\prime }}$ is a smooth vector field on $N$ with $\mathrm{supp}{X}_t^{\mathrm{\prime }}\subseteq \bar{V}$, so ${\bigcup }_{t\in I}\mathrm{supp}{X}_t^{\mathrm{\prime }}\subseteq \bar{V}$ is compact.

The ambient isotopy. Apply [F5] on the compact interval $I$ to the family $X_t^{\mathrm{\prime }}$ of step 2.1 and let ${\mathrm{\Psi }}_{t,s}$ be the resulting global evolution operator. Put $H_t:={\mathrm{\Psi }}_{t,0}$ for $t\in I$. Then $H_0={\mathrm{i}\mathrm{d}}_N$, the map $H\text{\hspace{0.17em}⁣}:N\times I\to N$ is smooth, and each $H_t$ is a diffeomorphism with inverse $H_t^{-1}={\mathrm{\Psi }}_{0,t}$, by the cocycle law in [F5]. Since $\mathrm{supp}{X}_t^{\mathrm{\prime }}\subseteq V$ for every $t$, every trajectory of $X^{\mathrm{\prime }}$ starting outside $V$ is constant, so $H_t={\mathrm{i}\mathrm{d}}_N$ on $N\setminus V$ for every $t$; a fortiori $H_t$ is the identity outside $W$, because $V\subseteq W$. Finally $X_t^{\mathrm{\prime }}=0$ whenever $t\le \frac{\epsilon }{2}$ or $t\ge 1-\frac{\epsilon }{2}$, because $\beta$ vanishes there. Uniqueness of the evolution with zero velocity gives ${\mathrm{\Psi }}_{t,s}={\mathrm{i}\mathrm{d}}_N$ when $s,t$ belong to either one of those intervals. Starting at time zero therefore gives $H_t={\mathrm{i}\mathrm{d}}_N$ on the initial interval. On the terminal interval the cocycle law gives $H_1={\mathrm{\Psi }}_{1,t}\circ H_t=H_t$, so the ambient isotopy is stationary at its final map there.

The isotopy identity. Fix $x\in M$ and consider $\gamma (t):=F_t(x)$. Then $\gamma (0)=F_0(x)$, and for every $t\in I$ the derivative is ${\gamma }^{\mathrm{\prime }}(t)={\mathrm{\partial }}_{t}F(x,t)$. Since $\gamma (t)\in K$ for all $t$, we have $\psi (\gamma (t))=1$, and therefore $X_t^{\mathrm{\prime }}(\gamma (t))=\beta (t){\mathrm{\partial }}_{t}F(x,t)\mathrm{.}$ If $t\in [\epsilon ,1-\epsilon ]$ then $\beta (t)=1$ and this equals ${\gamma }^{\mathrm{\prime }}(t)$; if $t\mathrm{\notin }[\epsilon ,1-\epsilon ]$ then either $\beta (t)=0$ or, by the standing hypothesis that $F$ is constant near the ends, ${\mathrm{\partial }}_{t}F(x,t)=0$; in both cases $X_t^{\mathrm{\prime }}(\gamma (t))=0={\gamma }^{\mathrm{\prime }}(t)$. Hence $\gamma$ solves the initial-value problem $\dot{\gamma }=X_t^{\mathrm{\prime }}(\gamma )$, $\gamma (0)=F_0(x)$, and by the defining property of the evolution operator in [F5] the unique solution is $t\mapsto {\mathrm{\Psi }}_{t,0}(F_0(x))=H_t(F_0(x))$.

Conclusion. Step 4.1 gives $H_t\circ F_0=F_t$ for every $t\in I$; step 3.1 gives that $H_0={\mathrm{i}\mathrm{d}}_N$, that every $H_t$ is a diffeomorphism, that $H_t={\mathrm{i}\mathrm{d}}_N$ outside $W$, and that $H_t={\mathrm{i}\mathrm{d}}_N$ for $t\in [0,\frac{\epsilon }{2}]$ and $H_t=H_1$ for $t\in [1-\frac{\epsilon }{2},1]$. Thus $H$ is an ambient isotopy of $N$ extending the isotopy $F$ of $F_0(M)$ and supported in the prescribed neighbourhood $W$.