Benedictines, historically, did not normally travel from place to place. The Rule of Benedict, the governing rule for that order of monks, included a rule of stability, which mean a dedication to remain at a single monastery. It appears that this was included in the Rule because of the problems in early monasticism with itinerant monks, whom Benedict (and his sources, e.g. Rule of the Master), found to be wanting; you can see this in chapter 1 of the Rule.

Anyway, the dominant early form of monasticism in Europe was benedictine, and this shaped the course of monasticism proper - apart from founding new monasteries, monks tended to be tied to a single place, though it was later relaxed.

In the 13th century due to a number of factors you get the emergence of mendicant orders, who are generally friars rather than monks. These groups, such as the Franciscans and Dominicans were founded without a vow of stability, indeed the expectation was that they would be itinerant travellers moving about the world in order to fulfil the goals of their order and thus the Church.

Lastly, the trope of video game monks using blunt weapons appears to derive from the classic D&D game, which in turn has the character class "Cleric" modelled generically on European priests. There is a canon law deriving from the medieval period in which priests are forbidden to shed blood, which I believe Gary Gygax took and turned into "blunt weapons only". The trope thus has its historical 'lineage', but it is not because of any tradition of warrior-monks or the like in the European medieval scene.

hi! lots of room for more responses here, but you may be interested in this post

• Were there any European Warrior monks besides the Knights of St. James and other military Crusader orders?