How to setup mysql replication with manual update option?

Question

MySQL

Hello!
There is a database (MySQL) to which it is necessary to provide access (by and large, read-only) from several client applications. Internet access for clients is unstable and may disappear for quite some time, but this should not affect the operation of the application. The solution that I see is database replication from the server to clients at times when there is an Internet and work on clients with their own local replica.
However, in rare cases, you may need "manual replication" (transferring dumps on a USB flash drive), in case the Internet is gone for a long time. As far as I understand, this can threaten to desynchronize the databases, since the position of the master log on the slave does not change, but the data is updated and the next time the slave will duplicate the same data during replication.
If, along with the database dump from the master, I save the current position of the master log, and then after applying the dump on the slave, I manually change its master log position to the saved one, will this solve the out of sync problem or not?
And what should I do with relay-log files on the slave? Will the slave not be confused by the absence of some records in the relay-log during the next replication?