So sorry for my (very) late response! I can't speak very broadly but I can speak to the particular incident of the Fourth Crusade, in which Venice played a very important part.

Innocent III, pope from 1198-1216, basically wanted a crusade to Jerusalem from the first day of his reign. The city had been taken by the Ayyubids in 1187, so for Lotario di Segni (Innocent) it was a fresh wound. He believed that the recent failed Crusades-- the Second, which got nowhere, and the Third, which gained ground but not Jerusalem-- would have been more successful if the pope had taken a stronger hand with them.

His early attempts to raise a voyage to Jerusalem failed. But around the turn of the thirteenth century he gained some ground. This would become known as the Fourth Crusade, which you might know ended in the disastrous sack of Constantinople, the further destruction of good relations between East and West, and a noticeable lack of Christian control in Jerusalem. How? The answer lies in part in Venice's role in the crusade.

In the years leading up to the Fourth Crusade, Venetians had been expelled from the city of Constantinople. All efforts to redress this by Venetian leaders went incredibly poorly. The quite-powerful city was humiliated and left to stew in it for decades. In 1204, Enrico Dandolo was doge in Venice. He had been personally involved with attempts to fix the troubles between Venice and Constantinople during decades earlier. He and his city were also the only ones capable of providing the ships that Innocent needed to send the *miles Christi* to Jerusalem. Famously, the overland journey was monstrously difficult during the First Crusades, and relations between the West and the Greeks had not improved much, which would further complicate things if they were to try to go by land. So they didn’t bother. Sailing to the Holy Land would be much simpler.

Innocent was aware of how the Venetians resented Constantinople but, lacking better options, had to content himself with them. He made all the leaders of the campaign swear that they would under no circumstances attack Christians on their journey.

As it turned out the campaign became much more complicated-- crusades always do. They ended up in Zara, a city in modern-day Croatia. They conquered it, despite the fact that it was Christian. In many ways this functioned as a foreshadowing of the sack of Constantinople. When they got to the city, crusaders tried to put Alexios IV Angelos, whose family was more sympathetic to the Franks, on the throne in place of his uncle. Alexios IV had promised to pay off debts owed by the crusaders to the Venetians, incidentally.

Well, things went south and the crusaders ended up sacking the city. The following destruction and destabilization of the Empire has been said to have been a turning point in its decline. For Venice, though, it was a major coup. They seized relics from the city and received a great deal of loot. Those relics could be used to bolster Venice’s image as a holy city in the years to come, allowing for the establishment of their image as a pious participant in the Fourth Crusade.

Having read scholars like Jonathan Riley-Smith (see “Crusading as an Act of Love,” a seminal text) I don’t want to say that cities like Venice participated in the Crusades for exclusively material gain. But in cases like the Fourth Crusade, it’s hard *not* to see an ulterior motive. They had been humiliated only a few decades earlier and ended up diverting the campaign away from its original landing point in Egypt. Even Innocent understood the dangers of letting a Venetian-affiliated crusade run wild and tried to extract oaths from them that were not followed. Crusading could be very lucrative, both in terms of the actual profits and in moral and spiritual myth-making.

For more, see:

Bird, Jessalynn Lea., Edward Peters, and James M. Powell. Crusade and Christendom : Annotated Documents in Translation from Innocent III to the Fall of Acre, 1187-1291 Philadelphia: University of Pennsylvania Press, n.d.

Moore, John C. (John Clare). Pope Innocent III (1160/61-1216) : to Root up and to Plant Leiden ;: Brill, n.d.

Perry, David M. Sacred Plunder : Venice and the Aftermath of the Fourth Crusade University Park, Pennsylvania: The Pennsylvania State University Press, n.d.