= Solution
The growth bound is $|F(t,x)|\le K(1+|x|)$. A <characteristic curve> through $(s,a)$ solves
$$
\boxed{\dot X(t;s,a)=F(t,X(t;s,a)),\qquad X(s;s,a)=a.}
$$
The $C^1$ field is locally Lipschitz in space, giving local existence and uniqueness. On either direction of a bounded time interval the <Gronwall inequality> gives $1+|X(t;s,a)|\le(1+|a|)e^{K|t-s|}$. Thus no trajectory can escape to infinity at finite time. On the resulting compact space-time region, local <ordinary differential equation> existence continues it, proving global definition for all $t\in\mathbb R$.
Uniqueness gives the flow composition law and inverse $X(s;t,\cdot)$. Differentiation with respect to $a$ solves the variational equation, making each map a $C^1$ <diffeomorphism>. Its positive <Jacobian determinant> is
$$
J(t;s,a)=\exp\left(\int_s^t\operatorname{div}F(\tau,X(\tau;s,a))\,d\tau\right).
$$
These facts constitute the <global characteristic flow under linear growth>. A uniform global bound on $D_xF$ is not required; its bounds on each compact characteristic tube suffice.
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