How often would monks leave their monasteries in the Middle Ages?

[This is in support of Luiz's answer, per comment, asking for sources]

1. On the Trappists: They are a spin-off of the Cistercian order. It is not surpising that Trappists do visitations (to other monasteries) and attend annual General Chapters. This is in fact a requirement of the Order of the Cistercian, based on their constitution, Carta Caritatis (Latin for Charter of Charity).
2. On monasteries having to be self-sufficient: The Carolingian monasteries of 8th & 9th century were highly dependent on land that they administer on behalf of the Carolingian overlord. See this.
3. On participating in important (political) meetings/councils: We should not be surprised at the milleu of monks during the High Middle Age, requiring them to work with kings and nobles. Recall the Investiture Controversy beginning with Pope Gregory VII and Emperor Henry IV of early 11th century. By early next century, 1122, this controversy was settled at Concordant of Worms - where it was agreed political and religious authority on investiture should be separate. Yet, less than a hundred years later, there was the Avignon Papacy lasting nearly 70 years (from 1309 to 1376). This was where Boniface VIII had to move away from Rome, to Avignon -- to be closer to the French crown.

And the list goes on ... Luiz covered so much in the answer. Hence, my applause for it being well thought out.

Typically, a significant minority of the monks were priests. But each order would have its own rules about the movement of its members. The more strict orders would not accept that one of his priests would serve the local village parish - e.g. the Trappists don't.

Even the most strict orders, such as the Trappists (see the movie Into Great Silence, which follows a year of their lives), who would regularly live their whole lives inside the monastery, still had exceptions, e.g., periodically senior monks would travel and visit other monasteries, to exchange experiences, detect eventual errors or deviations of the rule, and discuss potential changes or evolutions on their observance (see, e.g, the discussion in the movie about washing their hands before having lunch). I have never heard of a religious organization present in various cities (not only religious orders) that don't have some kind of periodic visitation to check their observance. The founding of new monasteries is another obvious exception.

Most other orders, even contemplative ones, would have larger exceptions. Many monasteries aimed to be self-sufficient, and thus had to administer lands worked by serfs, and/or their own business such as liquor, beer, or perfume making, and although they could employ middle men, the eye of the owner is often necessary - they would have to inspect and talk with their serfs, their suppliers and to the clients who bought their produce.

They also could travel to help others, e.g., the Cistercians in Alcobaça, Portugal, would often travel around at the request of nobles or bishops to spread good agricultural or water management techniques in the XIII c - they were experts, the monks from France had brought recent techniques, and their own lands were among the most productive of the kingdom. They even had to build a palisade around the monastery to keep vendors out during prayer and night time - their extensive business meant that a village of middle men, employees, suppliers and vendors developed around their formerly remote isolated monastery.

Famous monks such as St Thomas or St Bernard, could also travel at the request of nobles (even kings or popes) to provide advice or participate in important meetings such as councils.

Monks that would teach at seminars or university had to move out of the clausure for teaching duties or academic meetings or ceremonies. Although, in some cases, their clausure were physically inside the university, such as some buildings in the University of Coimbra that were true monasteries with internal clausures that housed professor-monks. Orders dedicated to health care or teaching would also have similar requirements.

And, some orders do not live fully at clausure at all, such as the mendicant orders. Franciscans and Dominicans were supposed to be outside the monastery to preach or serve the poor, even if they are professed religious with vows of chastity, poverty and obedience, in the same way as monks. Traveling around nearby cities was also common for them.

Plus, the military orders (Templars and Hospitalers) - true monks, with vows. They had to administer their extensive lands, perform military training, go to war, and perform other services to nobles (another way to get income, as waging war is very expensive), such as french Templars transporting gold around for the king of France. A lot of time outside the monastery.

The Mercedarians would collect money to pay the ransom of Christians taken as slaves in the Muslim lands. This means that they had to move around to preach and collect the money, and then go to North Africa to negotiate the payment and release of the slaves. Some of them offered themselves to stay in North Africa to support spiritually the Christian slaves, when the local Emir would allow that.

In short, although many monks would not often leave their monasteries, and for some orders would very very rarely be seen outside, seeing a monk was not a rare event at all, if you count all the different orders.

Another case would be monks that were elected as bishops. But they should not be considered exceptions, as when a monk accepts the election, he is supposed to be dismissed from his vows specifically related to his former vocation, and embrace his new vocation as bishop with its own rules.

Another very general observation: many of the orders cited above were created during the Middle Ages. This means that during the Early Middle Ages, when monks were mostly Benedictines, Agostinians, or hermits, many of the exceptions cited above would not apply. On the other hand, some of the newer orders (Cistercians, Trappists) were stricter than the original ones.