Suppose that $M$ is a closed manifold, and that $f_t$ is a path of Morse functions, $0\leq t\leq 1$. I know that I can "follow" the critical points $c_1,\dots,c_k$ of $f_0$, meaning that there are smooth curves $\gamma_i:[0,1]\to M$ such that $\gamma_i(0)=c_i$ and $\gamma_i(t)$ is critical for $f_t$ (this is the implicit function theorem applied to $(t,x)\mapsto df_t(x)$).
Now, suppose that $M$ is a closed manifold, and that $(f_t,g_t)$ is a path of Morse functions $f_t:M\to\mathbb{R}$ and Morse-Smale metrics $g_t:TM\otimes TM\to \mathbb{R}$, $0\leq t\leq 1$. Is there a way to "follow" the stable and unstable manifolds of the critical points $\gamma_i(t)$ for the gradient defined by $g_t(\mathrm{grad}\,f_t,\bullet)=-df_t(\bullet)$?
In the case where $(f_t,g_t)$ is given by $(\varphi_t^*f_0,\varphi_t^*g_0)$ where $\varphi_t$ is an isotopy of $M$, there is no problem showing that $W^u(\gamma_i(t))=\varphi^{-1}_t(W^u(c_i))$, and so to get an identification as wanted. But this is too strong because it implies for example that the flows of all the gradients are smoothly conjugated, so it doesn't hold in the genral case. So my question is: are there some references dealing with that problem in the general case? Thanks for your answers.