MediaWiki supports moving pages with all subpages. Wikimedia has this feature disabled for non-privileged users. I cannot trace the exact reason for this, because the only thing the config says is "for now", and this was done before mediawiki-config was managed in git.

I think using the "move subpages" option is the way to go there, rather than having page moves behave differently on Module pages than other namespaces. Also note that you may not actually want to move *all* subpages, it may be that Module:Foo/Bar makes sense there even if Module:Foo is moved elsewhere.

BTW, things like enwiki's {{documentation}} template have the same issue.

Ah well. :) Thank you for the quick and clear answers.