Doing part of my PhD remotely in Europe while working full time? my US company offered me the prospect of doing my PhD (in public health, public policy, international development or something similar) at a European university while retaining my employment. They would also pay for the PhD. My thesis would be based on a research project I was already going to pursue in my work. I already have a Masters in Economics. Would a university be okay if I only worked out of the university for the first and the 2nd and 3rd year was finishing my research in the USA? Thanks!
You would need to check that with both the university and with any supervisor/advisor you would have. I'm pretty sure that such relationships do occur, even if not common. But being less the norm, you need to work out the details. 

In some ways the supervisor might be the bigger block than the university. It might take a trip to Europe to finalize any such arrangements. But you first need to determine if they would even consider such a situation. The answers will vary.