The Essos city Tyrion was traveling toward when he met Young Griff in the novels is Meereen.
In A Dance with Dragons, Tyrion Lannister leaves Pentos under a false identity and travels with Griff and Griff's supposed son, Young Griff. Their stated destination is Meereen, where Daenerys Targaryen is ruling after conquering the city and much of Slaver's Bay. Tyrion gradually realizes that Griff is actually Jon Connington and that Young Griff claims to be Aegon Targaryen, the supposedly murdered son of Rhaegar Targaryen.
Tyrion's meeting with Young Griff is part of a major book storyline omitted from Game of Thrones. During the journey, Tyrion plays cyvasse with the young claimant and manipulates him into abandoning the plan to seek an alliance with Daenerys. Young Griff instead chooses to pursue his own invasion of Westeros.
Volantis is a common mix-up because Tyrion passes near it and is later abducted by Jorah Mormont in the region. Pentos is another tempting answer because Tyrion begins the journey there, but Meereen is the intended destination.