If any headquarters need to see any aggregated data or any client needs to see any cnage made by an other client - than you have no other choice but to sychronize. The what and how depends on the application and other requirements.
But as SQLite is an embedded RDBMS - it is serverless. So synchronize with what? You can't sync all to all. That's no way. You need a server, some server.
These might be interesting:
http://sqlite-sync.com/[
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http://atensoft.net/dev/sqlite3.php[
^]
Still, there are many things to consider - based on things you only can know in this situation...