Tyrion (not drowned; doesn't seem to have caught greyscale) learns some more from Young Griff aka Aegon Targaryen about how he came to survive. Varys bought a baby from a poor man and made a swap. Tyrion tells Griff he should just go to Dorne and take over Westeros himself. Once Dany hears about it she'll bring her dragons, he thinks, and this will be better than going begging to her.
Tyrion and Haldon the Halfmeister go off in search of news in Selhorys. Dothraki are lurking about in the wilderness. A red god preacher is preaching support for Dany being the chosen one. They go to the local bar and play cyvasse, which is chess but with cooler pieces. Tyrion loses on purpose but they get to hear some rumours about Dany. Apparently she feeds her dragons on newborn babies. It was one time! And by accident! C'mon, give her a break! Most dramatically they learn Dany is still holed up in Meereen and not heading for Westeros at all. This comes as quite a surprise to them.
Then Tyrion does the usual Tyrion thing of thinking with his dick. He visits a brothel, gets drunk, doesn't find Tysha, does find someone who recognises him and means to take his head to Cersei. So predictable, Tyrion.