Lie derivative "commutes" with pullback by time dependent diffeomorphisms

geometry differential-geometry manifolds riemannian-geometry smooth-manifolds

I don't think that anything in the equation depends on time varying. Or rather the equation is a relation at fixed arbitrary times $t=t_0$. So you just obtain a diffeomorphism $\varphi_{t_0}$ and vector field $W(t_0)$ as for your example with $F$ and $V$. The equation should be interpreted as a family of fixed time equations.

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