Probably the most notable traveling pope of the Middle Ages was Leo IX (r. 1049-54) who was first in a line of reforming popes who spearheaded the papal revolution of the late 11th-early 12th centuries. In order to emphasize his power over bishops, after he was elected he went on a long circuit of northern Italy, France, and Germany to make his presence known and push his reforming program. Sometimes called the “Pilgrim Pope,” he traveled in a grand procession with a large retinue to make him (and his power) seem imposing. Wherever he went he held synods—councils of bishops—and deposed those guilty of buying their office (simony), allowing secular lords to invest them with their church lands, or were married. Famously, at the synod of Rheims he had the bones of St. Remigius placed on the altar of a new church dedicated in his honor and demanded the attending bishops to swear on them that they were not guilty of these crimes.

Popes who followed in his wake also traveled, though the most famous reformer among them, Gregory VII (r. 1073-85) did so unwillingly. Matched in a face-off with Holy Roman Emperor Henry IV over papal v. imperial power, Gregory first traveled to northern Italy to confront Henry, whom he had excommunicated. Later, when Henry invaded Italy, Gregory was forced to flee in exile to Salerno (in S. Italy).

The reforming pope who launched the First Crusade, Urban II (r. 1088-99) traveled to a council in Clermont (700 miles north of Rome in France) to preach it in 1095.

The popes of the Avignon period (1309-77) left Rome altogether and moved the papacy to Avignon in southern France.

Whether the pope was a “popular religious figure” depends on when and for whom. Many of the popes of the 9th and 10th century were terrible, as in “facecube’s” account of Benedict and Boniface, and they were not the worst by a long shot. By the twelfth century, popes were much more respectable. Were the spiritually inspiring? That depends. The last medieval pope declared a saint was Celestine V who died in 1296. Did they leave a significant impact on the religious life of medieval Europe? Definitely, where Innocent III (r. 1198-1216) left the biggest footprint on the institutional development of the medieval church (and, for what it’s worth, was pretty pious too). Notably most of these later popes were not saints but were crack canon lawyers and theologians.

All things papal may be had in easy fashion in Eamon Duffy’s Saints and Sinners: A History of the Popes and J.N.D. Kelly’s useful Oxford Dictionary of Popes. A fun account of the "bad popes" of the Middle Ages is getting old (1969) but still lively: E. R. Chamberlin's The Bad Popes.