Dirichlet Process Mixtures and label switching

I'm reading a paper by Kurihara et al. "Collapsed Variational Dirichlet Process Mixture Models", 2007. The paper compares different representation of the DP prior and VB inference for these approaches. One of the flaws of the truncated stick-breaking representation in the VB setting is that it's not invariant to label switching, i.e. randomly permuting the labels changes the probability of the data.

In section "Optimal Cluster Label Reordering" the authors suggest a way to overcome this. Here's a quote:

The optimal relabelling of the clusters is given by the one that orders the cluster sizes in decreasing order.

I have a problem understanding two things about the proposed method:

1. How the relabelling procedure should be incorporated into the VB algorithm.
2. Why relabelling doesn't break anything.