Where was the center of the Roman Catholic Church located during the Middle Ages?

From 1309 until 1377, the pope resided in Avignon, France.  Before and after this time period, the pope resided in Rome.

Clement V was elected as pope.  He was French, and he refused to relocate to Rome.  He did not want to leave France.  He established the papal court in Avignon, France.  After Clement's death in 1314, six more popes resided in Avignon.  All of these popes were Frenchmen.  During this time, the pope and the French monarchy were closely aligned.

In 1377, the pope's residence moved back to Rome.  After this move back to Vatican City, there were tensions between French Catholics and Italian Catholics.  These tensions led to an unofficial split, as the French elected their own pope.  This new pope resided in Avignon, while the official pope continued to reside in Rome.