Relative Cohen-Macaulay modules under ring homomorphisms

Let $R$ be a commutative Noetherian ring with identity (not necessarily local) and $\mathfrak{a}$ a proper ideal of $R$. We study the invariance of some classes of $\mathfrak{a}$-relative Cohen-Macaulay modules under pure ring homomorphisms and ring homomorphisms of finite flat dimension. Our results extend several results in the existing literature on homological modules.