Smooth shifts along flows

Let $\mathrm{\Phi }$ be a flow on a smooth, compact, finite-dimensional manifold $M$. Consider the subsets $E(\mathrm{\Phi })$ and $D(\mathrm{\Phi })$ of $C^{\mathrm{\infty }}(M,M)$ consisting of smoothh mappings and diffeomorphisms (respectively) of $M$ preserving the foliation of the flow $\mathrm{\Phi }$. Let also $E_0(\mathrm{\Phi })$ and $D_0(\mathrm{\Phi })$ be the identity path components of $E(\mathrm{\Phi })$ and $D(\mathrm{\Phi })$ with compact-open topology. We prove that under mild conditions on fixed points of $\mathrm{\Phi }$ the inclusion $D_0(\mathrm{\Phi })\subset E_0(\mathrm{\Phi })$ is a homotopy equivalence and these spaces are either contractible or homotopically equivalent to the circle.