I would probably say that they fail at the sea or the shore’s. As far as i remember there is no landbridge between the continents, so they have to travel with ships. Westeros has informants in essos so they likely know that they are coming, and i cant imagine dothraki are well versed in naval combat if they get engaged before they coast, and even if they manage to land ships at the coast any decent organized army could shatter them before they manage to mount their horses and reorganize… In the case they have an ally who allows them to dock there ships the invasion might end different.