But there are many ways you can do that architecturally.
Here's one off the top of my head that might work:
1. Set up a DNS server on a home server.
2. Run a cron job to automatically wget every URL checked by the DNS server.
3. Set all of your devices to your DNS server.
4. Set up a local installation of one of the thousands of open source search engines and set it to go through all of your pages once an hour.
5. Enjoy.
Memex is not able to do cross-browser/device yet, but we are short before releasing our mobile apps for iOS and Android, as well as syncing between devices. Both the extension and mobile app will be offline first and sync happens with an end2end encryption via a relay server that deletes update message after every device has been picking it up.