I've not done this but it's intriguing; potentially a best-of-all-worlds solution.
I think "proper" automatic replication is not possible given the mismatch between Postgres and SQLite - not everything in Postgres maps to a thing that is possible in SQLite.
That said, there are a variety of ways to get data out of Postgres, and a variety of ways to get things into SQLite.
You could periodically export CSVs or whatever from Postgres and periodically import them into SQLite.
Or you could do a more realtime sync by using a Postgres foreign data wrapper like this one: https://github.com/pgspider/sqlite_fdw which would let you write directly to SQLite. Combine that with database triggers on the Postgres side of things and you've got something pretty close to realtime replication.
Those sorts of solutions wouldn't be as robust as "real" replication (specifically, what happens when one of the SQLite replicas is unavailable? do you catch things up later?) but could be very useful for a lot of scenarios. You could have Postgres importing gobs of data, and "replicating" it over to your read-only reporting server which uses SQLite as a data source.
My idea was to either use triggers or a process that reads Postgres's WAL and replay transactions to SQLite, by sending updates, rebuilding files entirely, or anything else.
Such replicator processes can be horizontally scaled, and made into an HA configuration.
Of course you should be careful to use on the Postgres side only the SQL features that will map well on SQLite. The range of such features is wide enough for practical applications though.
My real-life example would be a realty site, where changes are infrequent, but the desire to slice and dice the current large(ish) dataset in interesting ways is very high. The latter could be done using a per-node SQLite DB.
I think "proper" automatic replication is not possible given the mismatch between Postgres and SQLite - not everything in Postgres maps to a thing that is possible in SQLite.
That said, there are a variety of ways to get data out of Postgres, and a variety of ways to get things into SQLite.
You could periodically export CSVs or whatever from Postgres and periodically import them into SQLite.
Or you could do a more realtime sync by using a Postgres foreign data wrapper like this one: https://github.com/pgspider/sqlite_fdw which would let you write directly to SQLite. Combine that with database triggers on the Postgres side of things and you've got something pretty close to realtime replication.
Those sorts of solutions wouldn't be as robust as "real" replication (specifically, what happens when one of the SQLite replicas is unavailable? do you catch things up later?) but could be very useful for a lot of scenarios. You could have Postgres importing gobs of data, and "replicating" it over to your read-only reporting server which uses SQLite as a data source.