How can I compare classes from clusterings performed on two different data sets?
I suggest using Kullback–Leibler divergence (KLD) to compare the classes. The procedure is as follows:

1- Find the mean vector and covariance matrix of each class of each dataset.

2- Calculate the KLD between each class of the first dataset and all classes of the second dataset. So, for example if you have two dataset and each of them has 5 classes, you should calculate KLD, 25 times.

3- For each class of the first dataset, find the class from the second dataset with the lowest KLD between them. Doing this, you will find the most similar classes of the two datasets.

Note that KLD measures the similarity between two probability distributions and by calculating the mean vector and covariance matrix of each class, you are fitting a normal distribution to it. To find the formula for calculation of KLD between two multivariate normal distribution, refer to here:

https://en.m.wikipedia.org/wiki/Kullback%E2%80%93Leibler_divergence