What is the business requirement behind this request?
1 - Do you have some reference data in database 1 and want to replicate that data to database 2?
If so, then use cross database querying if you are in the same logical server. See my article on this for details.
2 - Can you have a duplicate copy of the database in a different region? If so, use active geo-replication to keep the database in sync. See my article on this for details.
3 - If you just need a couple tables replicated and the data volume is low, then just write a simple PowerShell program (workflow) to trickle load the target from the source.
Schedule the program in Azure Automation on a timing of your choice. I would use a flag to indicate which records have been replicated.
Place the insert into the target and update of the source flag in a transaction to guarantee consistency. This pattern is a row by agonizing row pattern.
You can even batch the records. Look into using the SQLBulkCopy in the system.data.sqlclient library of .Net.
4 - Last but not least, Azure SQL database now supports the OPENROWSET command. Unfortunately, this feature is a read only from blob storage file pattern when you are in the cloud. The older versions of the on premise command allows you to write to a file.
I hope these suggestions help.
Happy Coding.
John
The Crafty DBA