Although I can't say I know precisely what the average journey of a Norman would look like during the specific crusades, the presence of people originating from Normandy migrating Southern Italy (and Sicily) has a long, complex, and (I think) interesting history independent from and indeed preceeding the crusades.

In fact, the Normans (perhaps as a semi-conscious hark to their Viking forefathers) were particularly predisposed to offer their services as bodyguards for people traveling to the holy land long before the crusades. Of course, these early travelers would be a tickle of mostly clergymen and the most pious nobility, but it's an interesting precedent nonetheless. When these Normans invariably returned by passing through the southern half of Italy, they found the Principality of Salerno (a rump-state of the defunct Kingdom of Lombardy) and about a half dozen splinter Duchies locked in systemic conflict with both the Byzantine Empire and the Abbasid Caliphate. Soon, a small clique around four Norman brothers, with retinues that could be as small as thirty followers (at least according to the Byzantine chronicler Anna Comnenos) entered the permanent service of Guaimario, the Prince of Salerno. Before long, ethnic Normans were a leading component of the Salernitan armed forces, contributing to the stabilization of Guaimario's borders. The Norman leaders, brothers William, Roger, Drogo, and Robert, went on to carve for themselves a semi-autonomous fief in the Italian region of Puglia, which they had seized from the Byzantine Empire. When in 1047 Guaimario was assassinated on the orders of the a particularly rambunctious subject, it would be the Normans in Puglia to rush to Salerno in order to place Guaimario's son Gisulfo on the throne in place of the conspirators' proposed usurper.

The events after Guimario's death proved that the Normans were not just scale tippers, the were integral to Salerno's stability and strength. Not only had they undermined a usurpation plot, they subsequently carried on a campaign to expel the Byzantines from the province of Calabria. This might have contributed to Gisulfo's feeling they were getting out of control and attempt to isolate them. All would come to a head in 1053, when Gisulfo struck an alliance with the emergent Papal States to the north in order to organize a punitive expedition in Puglia. The outcome was a resounding defeat, causing Gisulfo to retrench himself in his capital. By 1076 the Normans had removed Gisulfo from power altogether.

The four Norman brothers would go on to conquer the whole of Sicily, paving the way for one of their descendants to be crowned King: a rather dramatic ascendance.

What I hope I was able to transmit is the way the Normans integrated themselves in the martial/land-based political hierarchy in Southern Italy, and once they built a power base for themselves they almost inevitably neutralized the disunited and dysfunctional political class. Indeed, Guaimario's vitality and effective ruling style, which must have played no small part in convincing the Normans to stay and fight in the South, was most certainly the exception rather than the rule. The prosperous, commercially vibrant, and agriculturaly fertile state in Southern Italy was politically isolated and had became inward looking to the point of being beset by dysfunction and endemic separatism.

If you'd like to read more about the Normans in Southern Italy, John Julius Norwich (my absolute favorite italianist-by-way-of-byzantiuim) wrote a history in two parts, titled The Other Conquest and The Kingdom in the Sun respectively. Norman Sicily is described, by the aforementioned Norwich, as a society based on "Unequal Coexistence;" an unique but unbalanced coexistence of ethnic Arabs, Greeks, and Normans.

Privileged administrative positions, as well as generous grants of land, were given to Normans, Provençals, and Lombards, immigrants all (more or less in that order of preference). Tensions between the existing Sicilian Greeks and these new "Latins" were easily overcome by professing adherence to the Church of Rome, only recently (in 1054) irreparably separated from the Church of the East in Constantinople. Indeed, Greek Christians had in great part welcomed the Normans, and played a key role in mediating between the upper class of Latin lords and the subject populations of Muslim serfs (according to Johns J., The Monreale Survey: Indigenes and invaders in medieval west Sicily). It's interesting to note how within a generation of the Norman conquest, parish censuses show how serfs whose parents have Arabic names take Greek names; a testament to the dominant "Greek Christian" culture in spite of the new Norman ruling class and the almost entirely Arab administrative bureaucracy, and more importantly, the key role of Greek Churchmen in converting the Muslim segments of the population to Christianity.

John Julius Norwich, in his The Normans in Sicily puts it this way: “Norman and Lombard, Greek and Saracen, Italian and Jew – Sicily had proved that for as long as they enjoyed an enlightened and impartial government, they could happily coexist; they could not coalesce.” Thus the Normans were a tangible but small and rapidly diluting cultural segment in medieval Sicily. Indeed, Sicily must have been quite the melting post: Greek Orthodox inhabitants of the island could be expected to be fluent in Arabic, while some Muslims could be native Greek speakers. The Arab-Greek language and culture seemed, at least initially, to have no problem mixing with that of the newly arrived "Latins."