Local colliding with live site domain

I’m managing a site with a .org domain, and I have a clone of the site set up with Local using the .local domain. The .local domain works great, and of course requires the hosts file entries pointing .local to localhost. The trouble is that the changes I make on the local files appear to change the site when I load up the .org domain. Scary!

This happens after I import the database from the live site, and update URLs back to .local using this method (editing wp-config):

https://codex.wordpress.org/Changing_The_Site_URL

After some poking and prodding, I’m able to get the .org site to show that in fact it’s not being changed—but the browser very persistently shows the local changes that should only appear on the .local domain. By “very persistently” I mean the site shows the local changes even on different devices, on different internet connections (and I swear to you, the server’s files haven’t changed).

What could possibly be causing this? Some sort of browser shared cache (like Chrome syncing site cache across devices)? This issue seems to persist across cache clears, too.

If this can’t be solved I may have to find another dev solution. Not knowing if the live site is displaying correctly is a real non-starter.

Example: here’s the under-construction version of the site on .org, on my phone:

If you load up the site yourself you’ll see the correct, current version, which should look marginally better at the moment.