Consider Freelancing SE as a hypothetical [for now!] example. Freelancing has been in beta for 7+ years. Given its narrow scope, it doesn't get many questions each year, and that number is shrinking. It may be desirable for a site like Freelancing to expand scope; e.g. to permit self-employment or small business questions that would not previously have been considered on-topic.
I expect for an SE site to expand the scope of what's on-topic beyond what was contemplated at launch, the community would hold discussions at the community meta site and ideally achieve some kind of consensus. (I was involved as a pro-tem moderator at Money SE early on and participated in many of these kinds of discussions.)
But what about when the scope expansion contemplated by a community is significant enough to require both a new name and new URL? Using the example of Freelancing SE, what if it decided to broaden to Self Employment, or more generally, Small Business? Stack Exchange community moderators cannot effect such a change. I expect the Stack Exchange community management team would help with a change like this — but what would the SE community management team need to see from the community and moderators in order to agree to such a change and effect the actual change?
I looked for precedent elsewhere in the Stack Exchange network where a major scope change involved both a site name change and site URL change and discovered that Audio SE, while in beta, had merged with the Video and Film Production proposal.
However, that isn't a great example, for two reasons: First, the change was proposed and seconded by the founders of Stack Overflow, rather than at the grassroots level. Second, the resulting site did indeed get a new name and URL, but eventually the combined AVP SE site split into two.
Are there other Stack Exchange sites that have expanded scope significantly where a new name and URL were involved? If so, what was the process these communities followed, and has that process changed over time? If not, what would/should that process look like today?