On controlling the papacy
The papacy was indeed controlled by several dinasties during the middle ages by several dinasties. But, the means of influencing it, the array of families who could possibly influence it and the consequences for influencing it evolved during the different periods the papacy went through.
We can affirm safely that the Teofilatus family had a very firm grip at the papal see during the period from circa 900 to the year 1048. That control was tirannical and absolute. The seat was kept warm with suitable low profile popes, with some noticeable exceptions, caused by the occasional outburst of that unruled conjunt of mobs that was Rome at those turbulent days.
The last pope of the family, Benedictus IX (who was elected when he was around 20-25 years old) showed so low qualities (he arrived to the point that he actually SOLD the papacy) that at the end of a very troubled reign he got expelled. We could say that it was the start of the decline of the time where the swords of a local faction could decide the election of the pope.
That period would die out completely at the election of Benedicto Gaetani (Bonifacius VIII, one of the greatest jurists and lawyers the world has seen... but actually a simoniac, power greedy pope). His defeat at his bid of universal power met with reality as he faced the wrath of the French king and was humiliated at the attack of the Colonna (a family he had defeated and looted only to enlarge the Gaetani family coffers)... an attack paid and directed from France. After that, the popes by the sheer influence and power of the French crown, moved to Avignon, and it could be said that they acted as the King of France's personal chapel.
So, in the period from 1048 to 1303, the attempts of influence should be directly proportional to the proximity of the dinasty trying to influence them. The closest to Rome the dinasty should have its quarters, the more probable their "strong arms" could influence the mobs in Rome. That kind of influence would slowly decrease with time. The power of money should slowly increase, though. Gold will slowly outweigh steel.
This first period was marked by the abundance of popes of the two most ferocious roman families, the Colonna and the Orsini.
The next period would be from 1303 to 1376. At that time, as I mentioned the papacy see was in Avignon, not in Rome, although the papal lands were still the traditional territories around Rome.
Gregorius VII (after a failed attempt by Urbanus V) managed to move back to Rome. But died shortly afterwards. The personality of his successor (whose only real flaws were nepotism... and an extremely wrathful personality, lacking the lust of riches, the thirst for women and the hunger for power, his predecessors had), and the dubious circumstances of his election (he was not even cardinal, and it was suggested that he had been only elected by the council of the cardinals, because of the fact he was italian... under pressure of the angry mobs just out of the election chambers), provoked the schism.
The main factions during this 'french' period of the papacy would have been the 'french' and the 'lemousin', detailing the zones of influence of this two groups, northern-central and southwestern France. The dinasties living in this places, and of course, the current French royal family, aided by money, would be the deciding factor of the elections of this tiem.
The third period would be the time of the Schism. The influences, facts and problems in this time, that would go from 1376 (Urbanus troubled election) to the council of Constanza in 1415, and it was only the sheer influence of the kings and major families of europe, disgusted by the divisions brought up by the schism that made it possible for the papacy to reunite, with the election of Martinus V (the starting pope at EU2, I think).
The shiftings of alliances, the alignations and everything else, made depicting the possible influences of the ruling european dinasties, over the papacies worth of a complete different whole article.
From there, money ruled. Nearly absolutely. Only the influence of the most powerful kings in Europe could somehow compensate it... mildly. Nearly all popes or the kings supporting them bought literally the elections, until the struggle in the XVI century between Carlos I of Spain (Karl V, emperor of the HRE) and François I of France. The conflict between this monarch marked a change in the whole spectre of european politics, and thus of those sorrounding the pope.
But, limiting us to the years from 1415 to the end of crusader kings, we could say that money would be the nearly only factor deciding a papal election.
(I hope this will help a bit... but if anybody feels that this, or a part of this, is inaccurate I would thank any advice... My objective is to help a degree by which the papacy could be influenced across the years of the game).