C-infinity convergence of a periodic map evaluated along a convergent family

Let $E$ be a finite-dimensional real normed space, $F$ a real normed space, $T,r>0$, $a,b\in\mathbb{R}$, and let $\Phi:\mathbb{R}\times E\to F$ be $C^\infty$ on $\mathbb{R}\times B(0,r)$, $T$-periodic in its first variable there, with $\Phi(\varphi,0)=0$. If $v:\mathbb{R}^2\to E$ satisfies $v(s,t)\to0$ in $C^\infty$, uniformly in $t$, as $s\to+\infty$, then also

This carries $C^\infty$ convergence through coordinates built along a periodic orbit, such as tube coordinates near a Reeb orbit.

Formalization Note $C^\infty$ convergence is TendstoZeroCInfty (smooth on some $(s_0,\infty)\times\mathbb{R}$, every derivative tends to $0$ uniformly in $t$). $E$ is a finite-dimensional real normed space and $F$ any real normed space; $\Phi$ is only assumed smooth and periodic on $\mathbb{R}\times B(0,r)$.