Let $E \to M$ be a vector bundle over a compact manifold $M$, equipped with time-dependent bundle metrics, compatible connections, and a Ricci flow background metric. Let $\Phi:E\times[0,T]\to E$ be a time-dependent fiber-preserving bundle map, so that $\Phi(w,t)\in E_x$ whenever $w\in E_x$, and assume $\Phi$ is locally Lipschitz in the fiber variable on compact subsets of $E$, uniformly for $t\in[0,T]$, and continuous in time. Let $\mathcal K \subset E$ be a closed fiberwise convex set that is invariant under parallel transport. Suppose $s: M \times [0,T] \to E$ is a smooth section satisfying

\begin{align*} \partial_t s = \Delta_t s + \Phi(s,t), \end{align*}

where $\Delta_t$ is the connection Laplacian determined by the time-$t$ background metric and compatible connection. Assume that for every $x \in M$, every solution of the fiber ODE

\begin{align*} \frac{d\sigma}{dt} = \Phi(\sigma,t), \qquad \sigma(t) \in E_x, \end{align*}

with initial time $t_0\in[0,T]$ and initial value $\sigma(t_0)\in\mathcal K_x$ remains in $\mathcal K_x$ for as long as it is defined inside $[0,T]$. If $s(x,0) \in \mathcal K_x$ for all $x \in M$, then $s(x,t) \in \mathcal K_x$ for all $x \in M$ and $t \in [0,T]$.