Do you seriously think that if, in say 30 years, substantial chunks of the tropics-to-tropics range of latitude have become too hot/dry for mass agriculture, that people will not be trying to migrate?
I haven't seen anyone claiming that there is currently mass migration due to climate change. But I haven't seen anyone claiming that previous climate change events (anthropogenic or not) were not accompanied by mass migration (even if the absolute numbers of people were smaller due to total population)