Finally one related to my dissertation (11th century power struggles between monastic orders and the Holy See).

I'll try to explain why there was a slew of German-born popes followed by a century of French-born popes, proving that the 'italification' (per your question) of the papacy did not take place, but was rather the standard in the years before and after the periods mentioned in your question.

Firstly, the five (four, really?) German-born popes (Clement II, Damasus II, Victor II, and the most influential 'German' pope Leo IX) from your question are problematic, as they are interrupted by short reins of Italian popes (Sylvester III). The quick succession of popes in this period and the background of the German popes has to do with the investiture controversy. Who rules Christendom and gets to appoint clergy members wielding worldly powers, the emperor of the HRE or the pope? Henry III made use of the rivalling three popes reigning in the year 1046 and called a council at Sutri to sort the matter out. When the dust settled Henry chose his personal confessor, Suidger, Bishop of his recently created See of Barberg, as pope. His first act was to crown Henry as emperor. The German background of these popes should be seen in the light of the investiture controversy and the resulting sway of German emperors over the papacy. It's an exception, really. For further reading the following keywords will go a long way on JSTOR: Gregorian Reforms / Investiture controversy / Leo IX / Gregory VII. Gilchrist, John (1970). "Was there a Gregorian reform movement in the eleventh century?". The Canadian Catholic Historical Association: Study Sessions. 37

Next, the French popes. Your question mentiones 1261, but exactly two centuries before that the Arles-born Nicholas II recognised that electing several popes a year under pressure from emperors and Roman aristocracy isn't great for stability. The college of cardinals was given the power to elect the pope at the Lateran synod of 1059, giving the curia and the vicar of Christ more independence from German Kaisers. Nicholas could be counted as a French pope too, but I digress. The first French(/Arles-)-born popes of the period you mentioned are Urban IV and Clement IV, whose reigns were marked by a rehashing of the investiture controversy, this time called the Wars of Guelphs and Ghibelines. The names were likely introduced to Italy during the reign of Frederick Barbarossa. When Frederick conducted military campaigns in Italy to expand imperial power there, his supporters became known as Ghibellines (Ghibellini). The Lombard League and its allies were defending the liberties of the cities against the Emperor's encroachments and became known as Guelphs (Guelfi). Basically, this is the conflict between Church and empire all over again, which some extra spice in the form of Louis IX of France and his brother Charles of Anjou on the side of the papacy. Gregory X is the next pope and even though he was born on the Italian peninsula, we'll call him French for convening the Second council of Lyon and being besties with Philipp III of France. Kidding aside: by the time of his election there were so many French cardinals installed by the previous two popes that the Italian and French delegations were deadlocked for three years in what would be the longest papal election ever. The best sources for the conclave at Viterbo are in German and Italian: Franchi, Il Conclave di Viterbo (1268 - 1271) e le sue origini (1993). Fischer, Kardinäle im Konklave: die lange Sedisvakantz der Jahre 1268 bis 1271 (2008).

Thirdly the concept of nation states, 'German' and 'French' popes in the medieval period is problematic for obvious reasons since these concepts don't exist yet. Even calling a pope 'Italian' is difficult with an independent Genoa, Naples belonging to France and the HRE barging through Northern Italy. I do get your contrast between pontiffs stemming from the aristocracy of the papal states versus 'outsiders', so I tried to answer your question as best I could, explaining the realpolitik of pressure from the German Holy Roman Emperors versus a college stacked with French cardinals.

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The run of Germans in the 11th Century and the French popes in the 13th and 14th reflect political influence by secular powers in those periods. And they were both, in a way, related ultimately to the six centuries of Italians, interrupted only by the elections of the Spaniards Callixtus III and his nephew Alexander VI (both members of the influential Borgia family) in the 15th century, and by the Dutch Adrian VI, who reigned for less than two years near the beginning of the Reformation. Get ready, because this isn’t going to be a short answer. We’re going to cover, to the best of my ability, the history of papal elections and the pope’s relationship with secular rulers for the past thousand years or so. Strap in.

Today, when the pope dies or resigns there is a highly formalized process for choosing his successor. The College of Cardinals are the highest ranking officials of the Church hierarchy. Even though many of them are posted in various positions all over the world, each of them is appointed to be the titular head of a church in the city of Rome. This officially makes them Roman clergy, thereby eligible to choose the Bishop of Rome. They lock themselves into the Sistine Chapel, where they discuss, pray, and vote by secret ballot. The process takes many rounds, often stretching several days in modern times, because the election is decided by a two thirds majority. If you were around to watch the news coverage during one of the more recent conclaves in 2005 or 2013, you may recall that the paper ballots are burned after each round of voting. The smoke coming up the chimney is supposed to be colored black if the vote resulted in no winner, and white to announce that a new pope has been chosen.

Such a process was not in place during the first thousand years of Christianity. We know precious little about how the earliest Bishops of Rome were chosen. It appears that, at least in some cases, both clergy and lay people (including emperors and kings) were involved in papal elections. Byzantine Emperors were involved for a time, as were Frankish kings. The popes were technically elected, but a powerful ruler might “nominate” a preferred candidate, or communicate their displeasure with an objectionable one. Beyond that, this period isn’t my speciality. What is clear is that as the papacy gradually became a more powerful institution, secular rulers sought to influence both who was chosen and how they ruled. The counts of Tusculum, powerful Italian nobles, wielded enormous influence over the papacy beginning in the early 10th century. They even managed to have three members of their family chosen as popes in the 11th century. The Holy Roman Emperor weighed in as well, essentially appointing a number of popes (thus the run of Germans, but also remember that the Holy Roman Empire of the High Middle Ages was much larger than Germany is today, and plenty of "Italian" popes were technically born in the empire).

Incidentally, if the idea of a secular ruler influencing papal elections sounds old fashioned and corrupt, you should be aware that a number of Catholic rulers claimed the right to veto an unacceptable candidate into the 20th century. Emperor Franz Joseph of Austria successfully vetoed Cardinal Rampolla, the leading candidate at the 1903 conclave, and Francisco Franco tried (but failed) to block the choice of Cardinal Montini as Pope Paul VI in 1963.

But I digress. The Investiture Controversy of the 11th century marked a new phase in the relationship between the pope and secular rulers. It's a bit before my area of specialization, so I don't feel comfortable getting too deeply into it, but suffice to say it began as a dispute between the pope and the Holy Roman Emperor over who had the right to appoint (or “invest”) church officials in the empire. The emperor had exercised this power in practice previously, but the pope claimed authority. The disagreement eventually evolved into the basic question of who held the higher power: the Church or the secular authorities. After decades of war in Germany, the pope and his supporters (mostly) won out. The pope had the right to appoint high church officials (bishops, archbishops, etc.), though the emperor and other secular powers had authority in purely secular matters.

In the aftermath of the Investiture Controversy, the pope’s power was near its peak. Urban II was powerful enough to call for a crusade of all Christians against the Muslims in 1095. But more importantly for this discussion, the emperor’s influence in choosing the next pope was greatly reduced. The papal bull In Nomine Domini of 1059 firmly established that the College of Cardinals was the sole electoral body responsible for choosing a new pope. That was before the Controversy really heated up, but the emperor’s failure to retain his privilege of investiture even in his own lands meant that he had little ability to force his will in Rome. The rules were further tightened in the 13th century, requiring the cardinals to remain secluded together until a new pope was chosen (this following a nearly three year interregnum from 1268-1271 during which three out of the original twenty cardinals died, the cardinals were given only bread and water, and ultimately the roof of the building where they were meeting was removed, before they finally came to a decision and elected Gregory X).