How do I distribute data from the *source database* to the *destination database* in `SQL`?
Replication is the right solution here, despite the comment above.  Simply copy the published table into a separate staging table at each subscriber.  Then, on a schedule of your choosing, copy the data from the staging table to wherever it needs to go in the subscriber database.

You can use SQL Server Transactional Replication for this, or just copy the tables on a schedule with SSIS or over Linked Server connections.  

If you use SSIS, your SSIS package you can use a Foreach loop to load multiple databases, or you can configure a separate SQL Agent job or job step for each target database, which would enable you to schedule them differently, and run just one.  A SQL Agent job step can specify the target connection string.