The claim is absolutely true. They were brought along the trunk trade (large ship trade) conducted by the Italian merchant states, notably Venice and Genoa who were given access to the Black Sea by the Byzantines at different intervals. And Venice also had the ability to transport slaves from their control of the Adriatic Sea. Genoa in particular was very active in the transport of slaves and other illicit materials during the end of the 12th and during the 13th Centuries as they had trade connections in the Crusader states and with the Saracens.

The Slavic slave trade came to a halt eventually with the Papal Embargo which made it a moral crime punishable by excommunication or fines by the individual Italian cities to transport potential Christian converts to Muslim territories. The rationality behind this being that Slavs, although mostly non-Christian, could become Christian and increase the size of the Christian flock. There was a major emphasis in Catholicism on individual salvation, and selling Slavs to Muslims both strengthened Muslim states and would not allow the Slavic slaves to be redeemed through the one true faith. The major trade powers of Genoa, Pisa and Venice then started to use African slaves, which not perceived to be of the same quality and fetching a lower price initially, came to dominate the slave trade in the Mediterranean and eventually Western Europe.

If you want to read more about this topic I highly recommend Stefan Stantchev's book Spiritual Rationality: Papal Embargo as a Cultural Practice which deals directly with Mediterranean trade and The Great Sea by David Abulafia. Also Horden and Purcell's The Corrupting Sea.