The Internet Archive Wayback Machine is great for scraping content and saving it, and also for serving said content through a new archive.org URL the latter part of which reproduces the latter part of the source URL.
Proposed is a high-level deal with the domain registers and authorities such that
an abandoned website can be scraped and also its URL redirected. I'm thinking that a similar user interface might be used as a t present with a timeline floating at the top of the page showing previous captures of that URL. But the original domain would also be "captured" and so the original link from 15 years ago would still work.
What about domains which continue in use but are radically changed in their content? Well in theory the same thing could happen, when you enter an obsolete URL instead of getting a 404 page (or worse a silent redirect to the home page) you get hopped to the archive scrape and timeline while still being at the original URL.
What about individual URLs which are still active but have changed radically? Well in that case it would be up to the site owner whether to allow the Archive to put a floating timeline along the top.
Perhaps some at least of this functionality could be done client-side. But I would still like to see obsolete domains captured and vested in the Archive so they can be reunited with their scraped content.