convex optimization – Under what condition can we prove $nabla_x min_y f(x,y)=nabla_x f(x,y^*)$ where $y^*=argmin_y f(x,y)$?

Let $f: mathbb R^ntimes mathbb R^mto mathbb R$ be a function. I wonder under what condition can we prove $nabla_x min_y f(x,y)=nabla_x f(x,y^*)$ where $y^*=argmin_y f(x,y)$. For example, when $f(x,y)$ is jointly convex in $(x,y)$, then it is known that the above property holds. However, such assumption might be too strong and unnecessary.

  1. If $f(x,y)$ is only convex in $y$, does the above property hold?
  2. If $f(x,y)$ is non-convex, does the above property hold?
  3. If Item 1 or 2 is false, can we relax the property so that $nabla_x min_y f(x,y)=0$ iff $nabla_x f(x,y^*)=0$?

It seems that the answer is not clear on the website, so I come here for help. Thank you very much!