Smooth finite point motions extend to disk isotopies

A compactly supported field along the tracks. The boundary margins $1-\parallel z_i(t){\parallel }_2$ are positive on the compact interval; when $n\ge 2$, the finitely many pairwise distances $\parallel z_i(t)-z_j(t){\parallel }_2$ are positive there as well. Let $\delta >0$ be a common lower bound for all boundary margins and, when present, pairwise distances. By [L1] with $r=\delta \mathrm{/}6<R=\delta \mathrm{/}3$ choose a smooth bump $\rho :{\mathbb{R}}^2\to [0,1]$ equal to $1$ on ${\bar{B}}_{\delta \mathrm{/}6}(0)$ with support in $B_{\delta \mathrm{/}3}(0)$, and define $X_t(x):={\sum }_{i=1}^n\rho (x-z_i(t))z_i^{\mathrm{\prime }}(t)(t\in \mathbb{R},x\in {\mathbb{R}}^2)\mathrm{.}$ Each term is smooth in $(t,x)$ and the sum is finite, so $X$ is a smooth time-dependent vector field on ${\mathbb{R}}^2$ over $\mathbb{R}$ in the sense of [L2]. For fixed $t$ the supports of the terms lie in pairwise disjoint balls $B_{\delta \mathrm{/}3}(z_i(t))$ when $n\ge 2$, and these balls lie in $\mathrm{int}{D}^2$ because each point stays at least $\delta$ from the boundary. Moreover $z_i^{\mathrm{\prime }}=0$ on $(-\mathrm{\infty },0]$ and $[1,\mathrm{\infty })$. Hence the union of the supports over $t\in [0,1]$ is a compact subset of $\mathrm{int}{D}^2$, and $X_t=0$ for $t\mathrm{\notin }[0,1]$.

The flow carries each marked point along its path. By [L3] and [L5] the field $X$ has a global evolution operator ${\mathrm{\Psi }}_{s,0}$ over $[0,1]$. Fix $i$ and let $\gamma (s):=z_i(s)$. At the point $\gamma (s)$ the $i$-th term of $X_s$ equals $z_i^{\mathrm{\prime }}(s)$ because $\rho (0)=1$, and every term with $j\mathrm{\ne }i$ vanishes there because $\parallel \gamma (s)-z_j(s){\parallel }_2\ge \delta >\delta \mathrm{/}3$ while the support of the $j$-th bump lies in $B_{\delta \mathrm{/}3}(z_j(s))$. Hence $X_s(\gamma (s))={\gamma }^{\mathrm{\prime }}(s)$, so $\gamma$ solves the ordinary differential equation of [L2] with $\gamma (0)=z_i(0)={\mathrm{\Psi }}_{0,0}(z_i(0))$, and the uniqueness clause [L4] gives ${\mathrm{\Psi }}_{s,0}(z_i(0))=z_i(s)$ for every $s\in [0,1]$.

The flow maps are boundary-fixed diffeomorphisms. Since $X=0$ outside a compact subset of $\mathrm{int}{D}^2$, the flow through an initial point of $\mathrm{\partial }{D}^2$ is constant, so ${\mathrm{\Psi }}_{s,0}$ fixes $\mathrm{\partial }{D}^2$ pointwise and maps $D^2$ onto itself; it is smooth, and its inverse is the flow map ${\mathrm{\Psi }}_{0,s}$ of the same field, so by [L6] each ${\mathrm{\Psi }}_{s,0}$ restricts to a homeomorphism of $D^2$ that is smooth with smooth inverse, that is, a diffeomorphism. Taking $s=0$ gives ${\mathrm{\Psi }}_{0,0}=\mathrm{id}$.

Conclusion. Setting $\mathrm{\Phi }(s,x):={\mathrm{\Psi }}_{s,0}(x)$ and restricting the first variable to $[0,1]$ gives, by steps 2.1 and 2.2, a smooth map $\mathrm{\Phi }:D^2\times [0,1]\to D^2$ with ${\mathrm{\Phi }}_0=\mathrm{id}$, all time maps boundary-fixed diffeomorphisms, and ${\mathrm{\Phi }}_s(z_i(0))=z_i(s)$ for every $i$ and $s$; the terminal map ${\mathrm{\Phi }}_1$ therefore carries the initial marked set onto the terminal one, and no other property of the flow is used.