From what I understand, duels were pretty ritualistic and structured, but that each culture would have had its own traditions and processes to make a legitimate duel.
But what happens when you have participants and settings of different and potentially conflicting systems?
For example, an English man insults a French man. What happens next? Does it matter whether it happened in a French or English context? What if insult was given in the Ottoman court, where duelling was frowned upon?
This is what seconds tended to be for. Part of a second's role was negotiating the terms of the meeting and ensuring that both parties were satisfied that it was fair. Fairness was one of the most important parts. If a duel is unfair, its outcome couldn't be honorable, and duels are about satisfying honor, not harming your opponent.
Not every dueling culture had seconds, of course. Street duels in German cities in the 16th century, for instance, followed a complicated ritual of escalation but were generally started and completed on the same day or night that the insult occurred. Any nearby public witnesses would function as arbiters, to an extent. French duels in the 16th century could be granted a field by the king, who would similarly serve as a judge and arbiter, but this practice ended after the duel of Guy Chabot Comte de Jarnac and Francois de Vivonne Seigneur de Chateigneraie in 1547, and had been fairly rare in the years before that in any case.
French duelists in 1578, at the duel of the mignons, brought two seconds apiece, but their role was in fighting each other, rather than negotiating the terms. It should be pointed out that one of the duelists in that duel was German, though he'd been living in Paris for some time by that point. This was a trendsetting duel that brought a new intensity to dueling in France, and led (in part) to the "dueling craze" of the early 17th century.
Generally, dueling practices would likely follow whatever local trends were in use, but those trends might be in a state of flux or confusion, and any duel had the potential to radically change local practices. Duels were not rational, even if they generally followed patterns, they were an expression of violent emotion that could sometimes violate the expected pattern. Another example from the duel of the mignons was that the primaries, Entregues and Quelus, fought with unmatched weapons; Quelus had only his rapier, and Entregues had a rapier and dagger. Of all the combatants, Entregues left the duel with the least injury, but then fled Paris to avoid the king's wrath.
In cultures with seconds, the seconds were expected to be trustworthy peers whose job it was to ensure fair and transparent proceedings. So if one participant was foreign and didn't understand the local conditions, his second would have to be someone who did, and could navigate that system. But note that these were very complicated trends, and it was possible that locals would not see foreigners as peers, and therefore their insults would be beneath notice. The inverse might also be true; a humorous non-duel took place between English fencing writer George Silver and Italian fencer Vincent Saviolo; Silver publicly baited Saviolo frequently, challenging him to fight to prove whose system of defense was superior. Saviolo cheerfully ignored these challenges.
None of this is universal, of course, and again, duels aren't rational, they are emotional, and so while we can observe trends, hardcoded rules are not what we should be looking for. So, ultimately, in cases like you're describing, the conditions would be set following local trends, the dictates of the participants, and all would be subject to the intensity of the emotion involved.
A couple books you might be interested in:
Francois Billacois, The Duel: Its Rise and Fall in Early Modern France.
Markku Peltonen, The Duel in Early Modern England