Near convergence, the means can change ever so slightly; however, the small change in means may not be enough to warrant an object being closer to another cluster than the one it's currently assigned to.  K-means stops when there is no change in the number of objects assigned to clusters -- i.e., change in cluster sample size.  Some algorithms may monitor the means, but the "vanilla-flavored" K-means algorithm simply monitors change in cluster sample size.  So the answer could likely be: "it depends on the K-means algorithm (software)."  Software houses can do anything they want during algorithm development - obviously. 

If an algorithm monitored change in means, it would need to establish some sort of convergence criterion like the norm of the delta between the centroid vector at iteration $t$ and the centroid vector at iteration $t-1$. 

Perhaps something like the following for $p$ dimensions

$||\delta||$<1E-4,   

where 

$\delta=\begin{bmatrix} \mu_1^{(t)} -  \mu_1^{(t-1)}  \\  \mu_2^{(t)} -  \mu_2^{(t-1)} \\ \cdots \\ \mu_p^{(t)} -  \mu_p^{(t-1)}  \end{bmatrix}$.

The convergence criteria above is used extensively in a lot of unsupervised clustering and manifold learning algorithms.  However, for the simple cartoon example of K-means clustering, there's no need to wait for low value of the norm for the above delta vector, since the sample sizes of clusters will stop changing before the above convergence criteria is met.  In addition, you would have to toy around with what the optimal convergence criteria is: 1E-4, 1E-8, 1E-12, etc.  Whereas, stopping when the cluster sample sizes stop changing is much less computationally expensive.