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We generalize some known results for limit groups over free groups andresidually free groups to limit groups over Droms RAAGs and residually DromsRAAGs, respectively. We show that limit groups over Droms RAAGs arefree-by-(torsion-free nilpotent). We prove that if $S$ is a full subdirectproduct of type $FP_s(\mathbb{Q})$ of limit groups over Droms RAAGs withtrivial center, then the projection of $S$ to the direct product of any $s$ ofthe limit groups over Droms RAAGs has finite index. Moreover, we compute thegrowth of homology groups and the volume gradients for limit groups over DromsRAAGs in any dimension and for finitely presented residually Droms RAAGs oftype $FP_m$ in dimensions up to $m$. In particular, this gives the values ofthe analytic $L^2$-Betti numbers of these groups in the respective dimensions.

On subdirect products of type $FP_n$ of limit groups over Droms RAAGs