This happened extremely frequently, so we could probably come up with endless examples, but I can give you one example that has coincidentally come up in something I’m researching at the moment. In 1213, the Duchess of Brittany, Alix, married the French nobleman Peter of Dreux. They had a child, John, in 1217, but Alix died in 1221 when John was only 4 years old.

“While the fact that John was but four years old secured for Peter the actual control of the duchy until his son came of age, he would rule as custodian rather than as duke. He might still issue acts as duke of Brittany…but they would have force only as long as his regency continued.” (Scourge of the Clergy, 28-29)

And that is precisely what Peter did - he used the title “Duke of Brittany” and issued documents with it. He wasn't the duke though, and he never really had been. He was only the consort of Alix, so if she was dead, what authority did he have? His vassals in Brittany, and neighbours like the King of France, usually they referred to his “ballum”, his "custody" in Latin (also the root of the word “bailiff”). But he remained in charge of Brittany until 1237, when John was almost 20. After that, John was duke in his own name. Peter had no authority there, so he wandered around a bit, getting involved involved in political intrigues at the French court, and he went on crusade several times - he died on the way back from Egypt in 1250.

The use of the word “ballum” shows that there really was no conceptual difference between a regent governing a kingdom and one governing a county or duchy. For example, while Peter was on crusade with the king of France (Louis IX), the crusader kingdom in Jerusalem had no resident king. The king was Conrad, the son of the Holy Roman Emperor Frederick II and Queen Isabella II of Jerusalem (who had died in childbirth) - but Conrad lived in Italy and never came to Jerusalem, either as a child or an adult. While he was still a child, Frederick claimed to be his regent, but Frederick was mostly absent from the kingdom too, so it was governed by the “bailli” (the medieval French word for bailiff, also derived from the Latin ballum). When Conrad was an adult, Jerusalem was still ruled by the bailli.

Louis IX had also been an underage king and his mother, Blanche of Castile, governed as his regent. But her government was also described as a “ballum”. Emperor Frederick had also inherited his titles as a child - I’m not sure what tiles his regents used, but every time I introduce a new person here, someone governed for them as a child too…like I said, it was extremely common!

So, there wasn’t really a word for “regent”. Our word “regent” is simply the Latin adjective “regens”, which means “ruling”. The person “ruling” on behalf of a child would likely be described as something else, such as a custodian or bailli, regardless of whether the child was a local baron or a king or an emperor.

My source for Peter and John of Brittany is Sidney Painter, Scourge of the Clergy: Peter of Dreux, Duke of Brittany (Johns Hopkins University Press, 1937)