Lie derivatives of sections of natural vector bundles

Time derivatives of pullbacks and push forwards along smooth curves of diffeomorphisms of sections of natural vector bundles are computed in terms of Lie derivatives along adapted non-autonomous vector fields by extending a key lemma of Mauhart and Michor [Arch. Math., Brno 28 (1992)]. There is also the analogous result about the first non-vanishing derivative of higher order.