That I know, there was no actual structured pan-european system. While the running of the affairs of state, rulers encountered a whole host of people; when the topic of marriage came up, a number of considerations were made, and a match with a viable candidate from the ruler's social network was attempted. Sometimes considerations were political, other times they were personal, and more often than you'd expect they were a little bit of both.

Now, it's fair to say that monarchies in medieval and early modern Italy was particularly unstructured; a structured courtship system might have developed elsewhere at some point for all I know. Individual rulers could also choose to be more or less intrusive on their children's romantic considerations; one ruler might have a spouse imposed on them, while his or her successor might choose his own partner.

The influence a monarch's parents might have on the choice of their children's spouse also depended on when the match was being made; Galeazzo Maria Sforza was twenty-two and unmarried in 1466 when his father, the Duke of Milan, dropped dead. His mother, Bianca Maria Visconti, had set up a match with Dorotea Gonzaga, the daughter of the Marquis of Mantua. Although Mantua was a small but influential state pivotal to the balance of power in Northeast Italy, Galeazzo Maria categorically opposed the match. Galeazzo (and his mother) had participated in the Milanese delegation to an ecclesiastic council in Mantua and probably knew Dorotea, Galeazzo did not like Dorotea much at all.

Why? Well, Galeazzo Maria had spent a year in France which had evidently been a character defining moment: he would from then on be smitten by all things French and francophile. Probably unable, or unwilling, to go so far as to to secure a marriage to a French princess, he instead settled for the Paris-raised daughter of the Duke of Savoy. Galeazzo also appointed his brother Ludovico ambassador to the French King; it would seem that his mother Bianca Maria could do nothing to stop his rampant francophilia. Although Galeazzo's father, the previous Duke Francesco Sforza, had late in his life acquiesced to his son's requests and opened up a tentative negotiation to formalize a marriage with a dependent of the King of France to be framed within a broader arrangement of an alliance, on his ascension Galeazzo Maria rushed to complete these arrangements while his mother, realizing that her son would not hear her council, rather than have her decisions as Duchess constantly undermined resolved to retire to the palace at Lodi, where she would spend the rest of her days.

Why was the Duchess Bianca Maria opposed to the match? The Savoyard state, which did not yet include the foothills of the Monferrat and was only a marginal (albeit large) border terretory encompassing the mountains between France and Italy, was placed squarely in the French sphere of influence. The Duchy's rulers spent more time in Paris than in their castles perched the Alpine passes, and even the Duchy's lowlands are hummocky and in no way comparable to the rich floodplain of the Po basin; to make matters worse, a gang of barons had even attempted to kidnap Galeazzo Maria as he was returning to Italy from France. Bianca Maria, who had actively assisted her husband in the government of Milan, was extremely upset about her son's willingness to throw political considerations to the wind and marry the daughter of such a politically unimportant state (Bona's father, Ludovico, was also not historically very warm with the Visconti or Sforza, another disincentive). However, Bianca Maria ultimately could do nothing to stop him.

So that gives you a very colorful example of how complicated a marriage could be, balancing personal and political considerations: Galeazzo Maria wanted to position his Duchy closer to the Kingdom of France, marry a Francophone, and avoid marrying Dorotea Gonzaga if he could help it. Interestingly, Galeazzo Maria engineered a match for his son Gian Galeazzo to the daughter of the King of Naples almost as soon as the boy was born; a testament to the fact that even if one ruler made sure he had input in his own marriage, he might have different plans for his children. So again, this reinforces the point that there really wasn't a codified procedure.