In the US, it's normal for people to move to a different state to attend university, and then from there often to a different state again. Something like 1% to 2% move each year to a different state [0], which means in 10 years more than 10% of people will have done that. Some cross state borders multiple times in their lives. This is what we want to achieve, because it fuels economic growth and removes obstacles to happiness. (It's also, in a way, a restoration of what we had pre-Nation-State, when people were basically free to settle anywhere they wanted; but that's beside the point). Making the process of moving across the continent as smooth as possible is a Good Thing.
I would also move a point on terminology:
> working in a different EU country than their own
People don't own countries, they are born into them. (This is not just a pedantic remark - I think it's important that we move on from XIX-century nationalistic terminology if we want to achieve progress.)
[0] https://www.mymovingreviews.com/move/how-often-and-why-ameri...