How to synchronize / replicate a file database?

Question

.NET

Greetings!
There is a small database (SQLite), access goes through the Entity Framework, C#.
The structure is quite simple, plus there is a field by which you can find the modification time of each record.
At the moment, clients do not write to the database and synchronization is carried out by copying according to the script. Who has more base, that and dad. Sometimes all this is cleaned and so on.
At one time, the base was chosen as
1. Clients for the most part have very poor Internet access or do not have it at all. They can rarely download the full version of the database; they also do not really need the latest changes.
2. Do not write to the database.
3. Make a sample of 30MB (intensive reading).
I would like to solve some of the problems.
1. Make replication to clients using the methods of the database itself.
2. Let customers write a little.
Google suggests:
1. Big brothers, put a full server on the client... plus it's a bit off.
2. Third-party programs for synchronization (not embedded)
3. NoSql. Not exactly.
You can rewrite everything in a fairly large framework: change EF, change the base.
This may not be such a unique problem. I would like to hear in general terms how it is customary to do this :)
Of course, you can use handles, add one more connection and do the update with handles.

Answer 1

And if you put dropbox (or analogues) on all computers and put the database file there? Let him do the synchronization :)
Alternatively, there is a native implementation of the OneDrive API

Answer 2

Has a solution been found for this issue?
Is it possible to look at the result? (Quite interesting and if there is one)