The boom/bust cycle seems to be concentrated largely in the North, where the winters are more severe, and which is the only place we hear of having large-scale die offs. I wouldn’t expect the same pattern to happen in the Reach or Dorne, or at least not to any severity.
Moreover, in premodern economies where the overwhelming majority of the workforce is in food production, the tricky thing is that consumers are also producers. So yes, immigrants increase the demand on the food supply, but they also increase production.