First you want to make sure your vector field itself is well-defined. What do you intend to do when $f_i=f_j$?
No matter what you choose, your vector field will not be continuous except for very special $f$, so following it will be hard to define, but not impossible.
Note the following nice fact: Everywhere but those places, your vector field has the property that when you follow it, $\max (f(x))$ decreases. If it still has that property elsewhere, then following it long enough should allow you to reach the minimum value of $f(x)$.
Now, what should you choose? Let's say the first two coordinates of $f(x)$ are equal, for simplicity. Then your vector field should probably be somewhere in between $(-1,0,0)$ and $(0,-1,0)$. Otherwise, following it would take you into a region where the vector field there immediately took you out of it. But we can strengthen that condition. Suppose infinitesimally decreasing $x_1$ makes $f_1$ fall faster than $f_2$, and vice versa. Then your vector field must cause $f_1$ and $f_2$ to still be equal. Otherwise, you would immediately hit a region where the change was different.
