The Byzantine Empire
In the end, Byzantium proved to be more distraction than ally to the crusading effort, though one can hardly blame the Byzantines for this. The Empire was, by Western standards, immensely rich and powerful. To the pious crusaders, the Greeks should have been compliant and generous helpers to the main effort. To the cynical crusaders, Byzantium was a dangerous ally that could be trusted only so far as it was under Latin control. For the Greeks, of course, the Westerners were hardly more than a dangerous wild card, a risk any time it was played but so powerful the temptation to play it was always present.
These attitudes drove Byzanto-Latin relations from the First right through the Fourth Crusades. Once Constantinople had fallen, instead of being an immense engine driving crusader success, it became just one more state in the complicated equation of the East. More often than not, instead of coming to the rescue, the Latin Empire of Byzantium needed to be propped up.
After Michael VIII Paleologus recovered the city and ended the Latin Empire, matters largely returned to their old footing, but the Ottoman Empire soon brought a major new factor into the relationship. Increasingly, Byzantium was seen by the West as a bulwark against the Turk, a bulwark in need of support. Crusading efforts were more and more directed against the Ottomans and eventually at direct support of Constantinople itself.
When that city fell in 1453, the language of crusade became defensive. The faithful were called out into armies, but now it was to defend eastern and central Europe against Ottoman expansion into Hungary and Austria. This theme dominated right through the 16th century and even into the 17th. By that time, though, Byzantium was no more and even Constantinople no longer existed, for it had become Istanbul and the Hagia Sophia had become a mosque.