Thank you for the answer, I see that this is not the right question. Anyway, I have in mind some examples, when the function has some number of components of the critical set and it seems stable by perturbations. So, I wondered what invariant would be responsible for that.

So, it follows that for arbitrary $f_{t}$ some component of $A$ intersects both $\mathbb{S}^{2}\times\{0\}$ and $\mathbb{S}^{2}\times\{1\}$, isn't it? And we may claim such "stability" for any critical point of non zero index? Thanks for the answer.

No, I didn't check sphere eversion, as I don't really understand it. Is it a counter-example? Anyway, if Q=(0,0,-1) and (P,0), (Q,1) belong to one and the same component of set A, it does work to me.