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 all of the delta vectors for each centroid vector at iteration $t$ and iteration $t-1$.
Perhaps something like the following for $K$ clusters (centroids) each having $p$ dimensions:
$||\Delta||$<1E-4,
where
$\Delta=\begin{bmatrix} \delta_{11} & \delta_{12} &\cdots& \delta_{1k} &\cdots &\delta_{1K} \\ \delta_{21} & \delta_{22} &\cdots & \delta_{2k} &\cdots & \delta_{2K} \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ \delta_{j1} & \delta_{j2} &\cdots &\delta_{jk} &\cdots & \delta_{jK} \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ \delta_{p1} & \delta_{p2}& \cdots & \delta_{pk} &\cdots & \delta_{pK} \\ \end{bmatrix}$,
where $\delta_{jk}$ is equal to the difference between $\mu_{jk}^{(t)}$ and $\mu_{jk}^{(t-1)}$, the mean difference of the $j$th feature within the $k$th centroid at iterations $t$ and $t-1$.
Convergence criteria like the one above is similar to what's employed for 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 a 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 kind of "adaptive" to the dataset and centroids used, occurs much earlier, and is therefore much less computationally expensive.