Finer control on relative sizes of iterated sumsets

Inspired by recent questions of Nathanson, we show that for any infinite abelian group $G$ and any integers $m_1,\dots ,m_H$, there exist finite subsets $A,B\subseteq G$ such that $\mathrm{\mid }hA\mathrm{\mid }-\mathrm{\mid }hB\mathrm{\mid }=m_h$ for each $1\le h\le H$. We also raise, and begin to address, questions about the smallest possible cardinalities and diameters of such sets $A,B$.