There were many different ways of doing this! First of all I should describe how it would have worked for everyday, non-merchant people.

In places like France or England where there was originally no permanent national court, if you were involved in a dispute, or you were the victim of a crime, you’d simply have to wait until the king’s justiciars showed up.

England was relatively centralized. The kingdom was divided into counties, then into shires governed by a sheriff, and then into hundreds, governed by a bailiff. But they didn’t have any judicial powers, so the king sent out “itinerant justices” who would wander around from hundred to hundred to hear legal cases. This is the origin of a "circuit court" - the justiciars would make a circuit around the country and hear cases wherever they were.

In France they had the same idea, except France was basically the opposite of centralized, at least until the 13th century. The king’s authority didn’t really extend very far outside of Paris, and the rest of France was ruled by dukes and counts who were effectively independent - the Duke of Normandy or the Count of Toulouse, for example, were independent in a way that English aristocrats (e.g. the Duke of Norfolk, etc) never were. So all of these counts or dukes sent out their own justices too. They all had their own laws and customs, so by the early modern period, France was a patchwork of hundreds of different legal codes, unlike England which just had one.

Rendering justice this way obviously only worked if the people involved could stay in one spot until the justiciars arrived. But if, as you mentioned, they were merchants who had arrived for a fair, they would have returned home before the justiciars arrived. The solutions for this are generally known as “lex mercatoria” in Latin or “law merchant” in England (an archaic of way of saying “merchant law”). Merchant law dealt with debts, stolen and lost merchandise, allowed merchants to do things that might otherwise be prohibited to the native inhabitants (collect interest on a loan, gamble), and settled criminal matters.

In England, merchant law was used by “piepowder” courts or “staple” courts. The jurors in the piepowder court were made up of fellow foreign merchants as well as the native inhabitants of the town or village where the fair was being held. The court was held “from day to day and hour to hour”, i.e. whenever it was needed, not on a specific date in the future. If witnesses needed to be called and both parties in the case were foreigners, the piepowder court needed only foreign witnesses; if one party was native and the other was a foreigner then both foreign and native witnesses would testify.

Unfortunately things were not so equitable for foreign merchants who lived full-time in a city but did not have the same rights and liberties as native citizens. Foreigners living and working in London, for example, would negotiate individual trade agreements giving them certain rights to stay in the city and travel elsewhere in England, but they would have to return home after a fixed amount of time, pay heavy taxes, and the types and amounts of goods they could trade would be limited. But the opportunity for lucrative trade outweighed any hypothetical legal difficulties.

For less-centralized France, the Champagne fair is probably a good case study. The Champage fair was the biggest and most important fair in Western Europe. Champagne was, technically, part of France, but in reality Champagne was independent, and in the 13th century the count of Champagne was also the King of Navarre, so he was extremely powerful. Towns like Troyes and Provins were on the route north from the cities of Italy and southern France, toward Flanders and the Netherlands and then on to England or the Hanseatic cities in the North Sea and Baltic. This made Champagne a major commercial centre and the "fair" was actually a year-round never-ending series of fairs.

There was a separate “law of the fairs” and special justiciars to deal with any issues that arose among the foreign merchants there. This law was probably similar to the “law merchant” used in the piepowder courts in England.

The "law of the fairs" also applied outside of Champagne, if the merchants were coming to the fairs or returning home from them. In 1243, Count Thibaut IV had to settle a dispute with the city of Piacenza in Italy, which had attacked some other Italian merchants who were on their way to Champagne. He reminded the citizens of Piacenza that he had also protected them in the past when they were harassed in Marseille. The citizens of Piacenza refused to restore the stolen goods of the other Italians so Thibaut banned them from coming to the next fair.

In the south of France and Italy, and also in the crusader states where there were lots of French and Italian merchant communities, there were also merchant courts. They had the added benefit of a continuous tradition of Roman law. Roman-based law regulated debts and defined “native” and “foreign” more clearly than in the north. The more urbanized environment made it easier to settle disputes, since no one had to travel very far to get to the nearest court, and unlike the sprawling, rural, feudal territories further north, each city was more or less an independent state.

I can tell you all about how this worked in the crusader states. There, the Italian neighbourhoods in each city had their own independent courts, but there was also a “burgess court” for all the merchants in general, and everyone else who wasn’t the nobility, which was a lot like a piepowder court in England. Since the population of the crusader states was so diverse, they had procedures for all the possible circumstances - what to do if a Greek merchant owed a debt to a Muslim, if an Armenian craftsman assaulted a Syrian, etc etc. If you were a Greek, and you accused a Muslim of owing you a debt, you needed to find two Muslim witnesses to back up your accusation. You couldn’t bring in fellow Greek witnesses because they would be more willing to lie for you. Of course. it wasn’t totally fair, because anyone from the Latin Catholic class of crusaders automatically had more rights than non-Latins.

Merchants in the urban centres of the south and the crusader states also had public notaries, a profession that didn’t spread to the rest of Europe until later. If merchants had non-criminal disputes (concluding contracts, settling debts, etc), they could use a notary. More serious matters still needed to be heard before the local court though.

Throughout the Mediterranean, including in the Byzantine Empire and Muslim states in North Africa, they also made use of the ancient Lex Rhodia, the Rhodian sea law. This law helped sailors establish financial partnerships, settle debts, set prices for merchandise, and so forth, and it helped them settle disputes in case of shipwrecks or lost goods. This didn’t necessarily apply to international disputes, but a ship’s company could certainly include sailors who came from different areas and spoke different languages, and they might also become foreigners themselves when sailing to a far-off territory.

So, there were lots of mechanisms in place to deal with foreigners, from ad-hoc courts in major commercial areas in England and France, to Roman-influenced urban courts and legal texts in the Mediterranean.

Sources:

Keechang Kim, Aliens in Medieval Law: The Origins of Modern Citizenship (Cambridge University Press, 2000)

Theodore Evergates, Feudal Society in Medieval France: Documents from the County of Champagne (University of Pennsylvania Press, 1993)

Joshua Prawer, Crusader Institutions (Oxford University Press, 1980, repr. Sandpiper Books, 1998)

Hassan Salih Khalilieh, Admiralty and Maritime Laws in the Mediterranean Sea (ca. 800-1050): The Kitaab Akriyat al-Sufun vis-a-vis the Nomos Rhodion Nautikos (Brill, 2006).