Sounds good and pretty forward for me. Just go.
You use a local sqlite db as "cache". To keep it simple, do not implement any logic about that into your apps normal code. Just use the local db. Then, separately, you code a synchronizer. That one checks for the online connection and synchronizes the the local sqlite database with a remote database, maybe mysql.
This should be perfectly fine for all applications that to not require immediate exchange of the data with other processes all the time.
There is one catch, though: the low performance of sqlite on bigger data sets. That is an issue with all single file database solutions. So this approach probably is only valid for small data sets in total, or if you can reduce the usage of the local database to only a part of the total data, maybe only the time critical stuff.
Another workaround might be to use joins over two separate databases, the local and the remote one. But such things really boost the complexity of code, so think thrice if that really is required.