We consider the way sets are dispersed by the action of stochastic flows derived from martingale fields. Under fairly general continuity and ellipticity conditions, the following dichotomy result is shown: any nontrivial connected set $\mathcal{X}$ either contracts to a point under the action of the flow, or its diameter grows linearly in time, with speed at least a positive deterministic constant $\gL$. The linear growth may further be identified (again, almost surely), with a much stronger behavior, which we call "ball-chasing'': if $\psi$ is any path with Lipschitz constant smaller than $\gL$, the ball of radius $\gep$ around $\psi(t)$ contains points of the image of $\mathcal{X}$ for an asymptotically positive fraction of times $t$. If the ball grows as the logarithm of time, there are individual points in $\mathcal{X}$ whose images eventually remain in the ball.