Loosely speaking, a gradient flow
$$ \dot{x}(t) = - \nabla E(x(t)),\quad x(0) = x_0 $$
says that the trajectory of $x$ is evolving in the direction of steepest descent of some functional $E$ (usually called an energy functional). Here I keep things $\textbf{general}$, and do not assume $x$ to be in Euclidean space (i.e., $\nabla$ may not be the Euclidean gradient).
The idea is captured in the following image:
Is it possible for a trajectory to reach a point in which there are multiple directions of steepest descent of the functional $E$, and if so which direction will it choose to go in?