Let $Y$ be a smooth manifold, let $I\subset\mathbb R$ be an open interval, let $s\in I$, and let $P\in\operatorname{Diff}^m(I\times Y)$ be a properly supported scalar differential operator of order $m\geq 1$. Write its principal symbol in the $D_t=-i\partial_t$ convention as $p_m(t,y;\tau,\eta)$, and assume that the coefficient of $\tau^m$ is elliptic on the conic region under consideration. Suppose that, on an open conic neighbourhood of $J\times V\subset I\times (T^*Y\setminus 0)$, where $J\Subset I$ is a compact interval containing $s$ and $V\subset T^*Y\setminus 0$ is open conic, the normalized characteristic polynomial factors as

\begin{align*} a_m(t,y,\eta)^{-1}p_m(t,y;\tau,\eta)=\prod_{j=1}^m(\tau-\lambda_j(t,y,\eta)), \end{align*}

where the functions $\lambda_j$ are real-valued, smooth, positively homogeneous of degree $1$ in $\eta$, and pairwise distinct.

After possibly shrinking $J$ and $V$, let $\Phi_j(t,s):V\to T^*Y\setminus 0$ be the projected Hamilton flow determined by the characteristic branch $\tau=\lambda_j(t,y,\eta)$, defined for $t\in J$. More explicitly, $\Phi_j(t,s)$ is obtained by lifting $(z,\zeta)\in V$ to $(s,z;\lambda_j(s,z,\zeta),\zeta)$, flowing in $T^*(I\times Y)$ along the Hamilton vector field of $q_j(t,y;\tau,\eta):=\tau-\lambda_j(t,y,\eta)$ until the base time is $t$, and projecting to $T^*Y\setminus 0$. Let

\begin{align*} \Lambda_j(t,s)=\{(y,\eta,z,-\zeta)\in T^*(Y\times Y)\setminus 0:(y,\eta)=\Phi_j(t,s)(z,\zeta),\ (z,\zeta)\in V\}. \end{align*}

Let $\gamma_su=(u|_{t=s},\partial_tu|_{t=s},\dots,\partial_t^{m-1}u|_{t=s})$ denote the Cauchy trace where defined. If $U(t,s):(\mathcal D'(Y))^m\to\mathcal D'(Y)$ is a properly supported microlocal Cauchy evolution for $Pu=0$ on the propagated region from $V$ over $J$, normalized by $\gamma_sU(s,s)=\operatorname{Id}$ modulo smoothing operators microlocally on $V$, then each component of the $1\times m$ Schwartz kernel vector of $U(t,s)$ is, microlocally over $J\times V$ and modulo a smooth kernel, a finite local sum of Fourier integral operator kernels associated with the canonical graphs $\Lambda_j(t,s)$, $1\leq j\leq m$.