Let $(M,\omega)$ be a compact symplectic manifold.
Given a Hamiltonian funciton $H:M\rightarrow \mathbb{R}$ one can defined its Hamiltonian flow $\Phi^H_t$ and we will have that given $p\in M$ then $\phi_t^H(p)\subset H^{-1}(p)$ since $\frac{d}{dt}H(\phi_t^H)=dH(X_H)=0$.
Now I am wondering what happens in the case that $H$ is a time-dependent Hamiltonian. That is suppose we have $H_t:M\rightarrow \mathbb{R}$, we can still define its Hamiltonian flow $\phi_t^{H_t}$. My question is what happens now to the level sets under the Hamiltonian flow. That is suppose that we have $c$ a regular value for all $t$ would we have that the flow $\phi_t^{H}$ takes $H_t^{-1}(c)$ into $H_{t'}^{-1}(c)$ for $t'\neq t$? Or is there another phenomen happening? I tried to prove this basically doing the same thing as in the previous case but here we don't have that $\frac{d}{dt}H_t(\phi_t^{H_t}(p))=0$, at least I don't think we do, hence I am not sure what to do. Any insight is appreciated, thanks in advance.