Wasn't the church pretty much independent of the individual European kingdoms and empires? Also starting with the renaissance, but probably even before that, scientists and philosophers could move between courts.
The Church had its own religious purview but it took a while for it to become its own political force. In that respect it rose at the same time as the improvements in government, land management, international travel, trade etc. that occurred in the later half of the Middle Ages.
The first comprehensive system of intellectual dissemination came through universities, starting in the 11th/12th c. Those who wanted to study law went to Bologna, theology to Paris, medicine to Montpellier etc. Universities were international places where people used Latin as a vehicular language. Broke student spending their time drinking and living in hovels were legion and became a hotbed of radicalism [1].
By the time the Renaissance came around, there was an entire industry of private education with lay academies everywhere and teachers travelling all over the continent.
