Congratulating professor to having moved to a new university? I want to get back in touch with a former professor of mine, who has now recently moved to a different institution. Would I, in my email, congratulate her for having moved to the new institution or should I just not mention it at all? 

I am worried it could be understood the wrong way.

For what it matters: both the old &amp; the new university are top 20 in the world, the new university is top 3 (in the relevant field).
If your relationship was all one-way, as professor-student relationships often are, then I would not mention it.

If you sometimes had casual conversations that made things more two-way, and you happen to know something that provides context to this move (for example, if this is a solution to a two-body problem, or you happen to know that the professor was hoping to move to a certain geographical region, e.g. to get in more skiing or hiking or snorkeling, or to support an aging relative) then sure, you could include congratulations.