Compactness gives a compactly supported time-dependent velocity field

A relatively compact neighbourhood of the track inside $W\times I$: $S$ is compact by [F1] and $S\subseteq F(M\times I)\times I\subseteq W\times I$, which is open. Since $N\times I$ is locally compact Hausdorff and $S$ is compact, [L2] applied at each point of $S$ yields finitely many open sets with compact closure covering $S$ and contained in $W\times I$; their union ${\mathrm{\Omega }}_0$ is an open neighbourhood of $S$ with $\overline{{\mathrm{\Omega }}_0}$ compact and $\overline{{\mathrm{\Omega }}_0}\subseteq W\times I$.

Choose a smooth cutoff $\psi :N\to [0,1]$ with $\psi =1$ on the compact image $F(M\times I)$ and $\mathrm{supp}\psi \subseteq W$ by [L3], applied to the closed set $F(M\times I)$ inside the open set $W$; and choose a smooth function $\beta :I\to [0,1]$ with $\beta =1$ on $[\epsilon ,1-\epsilon ]$ and $\beta =0$ on $[0,\epsilon \mathrm{/}2]\cup [1-\epsilon \mathrm{/}2,1]$, which exists by the smooth Urysohn lemma on the interval.

Apply [L1] with the prescribed neighbourhood ${\mathrm{\Omega }}_0$: there is an open neighbourhood $\mathrm{\Omega }\subseteq {\mathrm{\Omega }}_0$ of $S$ and a smooth horizontal $\tilde{Y}:\mathrm{\Omega }\to TN$ extending $Y$. By [L3] applied to the closed set $S$ inside the open set $\mathrm{\Omega }$, choose a smooth cutoff $\rho :N\times I\to [0,1]$ with $\rho =1$ on a neighbourhood of $S$ and $\mathrm{supp}\rho \subseteq \mathrm{\Omega }$.

Define $G$ on $\mathrm{\Omega }$ by $G(y,t):=\rho (y,t)\beta (t)\psi (y)\tilde{Y}(y,t)$, and define $G:=0$ on the complement of the closed set $\mathrm{supp}(\rho \beta \psi )\subseteq \mathrm{\Omega }$. The two definitions agree on the overlap, where $\rho \beta \psi =0$, so $G$ is a well-defined smooth map on all of $N\times I$: at a point outside $\mathrm{supp}(\rho \beta \psi )$ it vanishes on a whole neighbourhood, and on $\mathrm{\Omega }$ it is a product of smooth functions with the smooth map $\tilde{Y}$. It lies in $T_{y}N$ at $(y,t)$ by construction. Thus $H:I\times N\to TN$, $H(t,y):=G(y,t)$, is smooth by composition with the factor-swap map and has $H(t,y)\in T_{y}N$, so it is a time-dependent vector field on $N$ by [L4], with slices $H_t=G_t$.

Clause 2: let $(x,t)\in M\times I$. Since $\bar{F}(x,t)\in S$ and $\rho =1$ near $S$ and $\psi =1$ on $F(M\times I)$, one has $\rho \beta \psi \tilde{Y}(\bar{F}(x,t))=\beta (t){\mathrm{\partial }}_{t}F(x,t)$. The isotopy is constant near the ends, so ${\mathrm{\partial }}_{t}F(x,t)=0$ for $t\le \epsilon$ and for $t\ge 1-\epsilon$, while $\beta =1$ on $[\epsilon ,1-\epsilon ]$; in all cases $\beta (t){\mathrm{\partial }}_{t}F(x,t)={\mathrm{\partial }}_{t}F(x,t)$. Hence $G_t(F(x,t))={\mathrm{\partial }}_{t}F(x,t)$.

Clause 3: for $t\le \epsilon \mathrm{/}2$ and for $t\ge 1-\epsilon \mathrm{/}2$ one has $\beta (t)=0$, so $G_t=\rho \beta \psi \tilde{Y}(\cdot ,t)=0$ by step 3.1.

Put $K:={\mathrm{pr}}_N(\mathrm{supp}\rho )$. The support of $\rho$ is closed and contained in the compact set $\overline{{\mathrm{\Omega }}_0}$, so it is compact, and its continuous projection $K$ is compact and contained in $W$. Off $K$ the field $G_t$ vanishes for every $t$. Since $K$ is closed, every $\mathrm{supp}{G}_t$ lies in $K$. Thus their union and its closure lie in the compact subset $K\subseteq W$, proving clause 1. Containment alone would not prove that the union itself is closed.

Clauses 1, 2 and 3 are steps 4.3, 4.1 and 4.2; the field is smooth and horizontal, and countable choice was used only as declared in [A1].