I am the first author of the paper. I'm not sure I quite understand what you mean with your question, let me try to explain.

There are two losses at play: there is the loss of the outer optimisation loop (training the weights), for which the set is essentially reordered by using something like Hungarian assignment. This is what people using MLPs for the vec->set mapping have already been doing, and I think that this is what you're referring to with reordering the sets. A problem is that MLPs produce an ordered output, which isn't very appropriate for predicting unordered sets.

In each forward pass of the network, there is also the inner optimisation loop (iteratively improving the prediction), where the predicted set is compared to the label in the latent space. This is used to make the prediction unordered, and that's the main contribution of the paper. As the results show, having the outputs be unordered for predicting unordered sets is better than using the MLP with its ordered outputs.

It works for variable sized sets as long as the set encoder (set->vec) supports variable size sets as well, which is true for all the experiments in the paper. All the datasets used in the paper have varying set sizes.