Would one, for instance, be subjected to pretty much the same mix of intellectual influences and opinions in a Franciscan monastary in Ireland than in a Cistercian one in Italy, of would each monastary have it's own "personality", for lack of a better term?
So just to clear up some things: the Cistercians are "founded" in the last years of the eleventh century, and the Franciscans are the product of the thirteenth, so we can talk about neither in the ninth.
In the Merovingian period, there were a profusion of available rules. These rules were proscriptive texts, usually named after the holy man purported to have written them, with the Rule of Columbanus being one of the more popular. Monasteries were founded "under" these rules, or under some union of several of them, but it's really not clear what this means in practice. There's been a lot of work done recently that's come to the conclusion that our current idea, that the rules were akin to local law, overstates their importance. A rule might form a point of cohesion and give a community grounding, but they were frequently more like guidelines. Moreover, two monasteries claiming to follow the same rule is not a guarantor of orthopraxy, or even similarity, between them, nor did it imply a sense of community between them - that is, they were not an order.
Under the reforms implemented by Benedict of Aniane, part of the broader Carolingian church reforms, we do see a growing unity of monasteries under the Rule of St. Benedict (of Nursia). From this point on, scholars are much more comfortable talking about "Benedictine" monasticism, a shared sense between monks of different monasteries that they were part of the same group that followed the same rule. Still, before Cluny arose in the tenth century, there was no real monastic order because there was no overarching structure. And large degrees of local variation, recorded for us in monastic customaries, persist within monasteries under Benedict's Rule, even to the present day. These include differences in the ordering of the monastic day, in the liturgy, in which feasts are observed, etc. Even the Cluniac structure imposed only a basic level of orthopraxy between houses; liturgical uniformity and the like would have to wait several more centuries.
So yes, each monastery would be fairly distinct.
There isn't much that's accessible and recent on the subject, but Isabelle Cochline and Alison Beech should have their Cambridge History of Monasticism out any day now (I was told a year ago) through Cambridge University Press, so when that comes out, go there.