Given two vector fields $F_1,F_{2}: \mathbb{R}^n \rightarrow \mathbb{R}^n$ with their flows $\phi_{1}, \phi_{2}:\mathbb{R} \times \mathbb{R}^{n}\rightarrow \mathbb{R}^n$
We say they are topological conjugate if there exists a homeomorphism $h:\mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ such that $$\phi_{1}(t,h(x))=h(\phi_{2}(t,x)) \quad \forall x \in \mathbb{R}^{n},\forall t \in \mathbb{R}$$ If we would treat the vector fields as maps $$x_{t}=F(x_{t-1})$$ we would call them topological conjugate if there is a homeomorphism $h':\mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ $$F_{1}={h'}^{-1} \circ F_{2} \circ h' \quad \forall x \in \mathbb{R}^n$$
My question is the following: Having a homeomorphism $h'$ for the vector fields. Can we lift it to a homeomorphism for the flows? What is the exact reason why $h'$ fails to be a homeomorphism for the flows even though it is one for the vector fields treated as maps?