The knight is taking Tyrion to Volantis, which has three parts. Old Volantis is poor and run down. Volantis east of the river is richer, more inhabited and pretty strictly Volantian. They pass the red priest, Bernello, who is calling Daenerys Azor Ahai reborn and saying that when she comes west the slaves should rise up to support her.
They stop to buy some gloves, put Tyrion in chains, and to sell the horse. Then they cross the longbridge to Volantis west of the river, where traders from all over the world come to do business. The knight heads for the Merchant House where he takes a room. He leaves Tyrion chained up. When he comes back he has food and ale.
The next day they go to have an audience with the Widow of the Waterfront, Vogarro’s whore, to whom the knight gives the gloves. She tells them things. He asks for a passage east. She tells them the Golden Company is going west to Westeros and asks why they want to go east. The knight says he’s taking Tyrion to Daenerys. Tyrion is overjoyed. The arrangements are agreed.