Does a PhD give you global mobility in the job market?
There are two aspects to this question: the hiring process within an employer's institution, and the immigration policies which allow a foreign worker to move to another country and begin employment.

I think with respect to the first question—the hiring process and preferences of an individual employer—I think that in general, employers are looking for the best-qualified candidates, regardless of their origin. There are certain obvious exceptions to this (for instance, a foreign national will have a hard time getting a position at a National Nuclear Security Administration facility in the US), but otherwise, people will want the person best-suited for the job. I think this is true both in the US and Europe.

The other step is the national immigration policies. This is where things become more complex. For instance, if you are a non-European Union citizen, then your employer will need to certify that your experiences uniquely qualify you for this position, which can delay the hiring process somewhat. Similarly, employers in the US have to wait for you to obtain an appropriate visa, which can delay your start date significantly. This makes it sometimes less advantageous to hire foreign employees. (But this is a "speed bump," rather than an insurmountable obstacle.)