The members of the 4th Crusades, impatient with waiting in Constantinople to head off to re-take the Holy Land from the Muslim hordes, after four years, instead attacked their hosts and sacked Constantinople. The English and the Germans grabbed the wealth, the French took the sacred relics. Later, the Knights Templar, when they were destroyed by King Philip the Fair (anything but fair) and the anti-Pope in Avignon, the remnants of the Templars went into hiding... and the treasures of the Templars were never found. One strange coincidence gives us a clue, from an early post of mine in this thread:
King Phillip order four men executed on March 14 (1314): Templars Jacques de Molay, Geoffroi de Charnay (Grand Preceptor of Normandy), Hugh de Peyraud (Visitor-General), and Guy d’Auvergne (Grand Preceptor of France). 40 years later, the Shroud would be put on display in a little wooden chapel in Lirey, France, by one Geofrey de Charney, King John II's Standard bearer, author of the French Code Of Chivalry, and thought to be either de Charnay's great nephew or grandson. The differences in spelling in those times was a matter of opinion. So there certainly appears to be some connection to the Templars.
I think that makes a very strong connection...
The Venetians also did some looting from their former Byzantine overlords, e.g. the Horses of Saint Mark.