Keeping many databases in sync all the time

I have a network of desktop PCs (Windows 7) which are located geographically apart from each other (connected with LAN).

The network has an Oracle back-end.

I want to install a database locally on each PC (about 12 of them, currently thinking of SQLite but open to other possibilities).

I need to guarantee the local databases are kept in sync with each other and with the Oracle db all the time or at least as long as there is network connectivity (mesh topology).

The synchronization involves only a single table.

What are some possible effective solutions for this problem ? preferably something you have worked with beforehand.