For me the most interesting point is the argument that across mammals (and perhaps other vertebrates!) the disruption of development is due singularly to changes in neural crest cells, but on the genetic level the evolutionary process is polygenic and diverse. In other words the developmental pathway will exhibit similarities, ergo, similar correlated side effect traits. But the genetic architecture of the change across species may vary, because there are many genes which are effected by the phenotypic target of selection. Another way to state this is that there is no gene for domestication in the lineages under consideration, but rather many genes which have significant, but not overwhelming, effect. Of course there’s polygenic, and then there’s polygenic. One of the common side effects of domestication is depigmentation of the pelage of mammals, but this is one case where the number of genes effecting the trait is relatively low, on the order of ten genes account for more than half the variation. In contrast you have polygenic traits like height where you’re lucky to find one locus which can explain one percent of the variation. If domestication is like the latter then the role of standing variation in the evolutionary story is going to be large, nearly total. In contrast if pigmentation is representative than classical selection on new mutations of large effect unique to particular lineages may still be important. Not to be lame, but the answer is probably going to be in the middle, on average.
Second, there are broader questions about contingency, the genetic architecture of salient traits, and selection as a driver for adaptation, which come to mind after reading this paper. It seems hard to deny that if you constrain the phylogenetic space enough then there are many instances where evolutionary forces will basically result in broadly similar phenotypic and genetic outcomes. Though there are some differences in traits and genetic variations, there is a great deal of overlap across mammalian taxa which have been targeted by artificial selection. Though the authors don’t address this directly it, seems clear that many of the phenomena which revolve around domestication also apply to humans. If they do, and if “domestication” occurs through gradual selection upon standing variation, then the search for the gene which makes us uniquely human (e.g., “the language gene”) may be futile. Rather than a gene, our humanity may have emerged out of gradual change as the underlying frequency of alleles is shifted. This is not a sexy answer which will result in genomic fame for a researcher who discovers the gene-which-makes-us-human. Finally, there is the issue where we bracket artificial selection and domestication as if they are unique processes which derive from human agency. My own position is that though for semantic purposes we may speak of ‘artificial selection’,’ sexual selection’, and ‘natural selection,’ there’s really no fundamental difference at the root for these phenomena. Selection is selection, and the rest is commentary. To me that implies that attempting to understanding domestication may actually allow us to understand evolution more broadly (and Charles Darwin would agree with that point I suspect).