Read some widely cited research on healthcare management, maybe starting at scholar.google.com Narrow your interest down to a specific topic. See
(1) which universities employ the authors of the most cited papers on your topic
(2) are there any people in your own country who write on this topic, perhaps co-authoring with any of them.

In some EU countries you will get paid to do PhD and gain permanent visa in the process, since you are hired by the university (you will probably get to do some 20hours/week TA to justify the salary - a good thing)

for example:
https://www.migrationsverket.se/English/Private-individuals/Studying-in-Sweden/Universities-and-university-college/Permanent-residence-permits-for-doctoral-students.html

Many phd projects are in English, and most schools have english programs where you can teach

That (stipend and visa) should really help with the relocation part!

For the teaching part... In my opinion the reputation of the university is very important and will follow you for years. So aim high

That's also the case for almost all PhD students at non-profit universities in North America, as well as in the Nordic and Benelux countries.
The visa path might differ a lot from country to county.
But, really, in the academic world what matters is where your research gets published, not the name of the reputation of the school. In this respect, see the UT Dallas ranking of business schools by research.