What you are looking to do is actually fairly easy to do yourself. What I would do is use a combinations of Azure Function and Azure SQL Integrated Change Tracking. Integrated Change Tracking gives you a way to ask the Azure SQL Database to tell you which rows have been inserted, updated or deleted. From this you can query these rows to then have them be sent to the other database. Using Azure Functions you can create the code to get these changes and then send them for you. The added ability of Azure Functions to do scheduling is also really helpful here.
Since you specifically asked about solutions that do this, I will mention the service I work on called Cotega and I am happy to help you do this, but once again with your requirements you might find it fairly easy to build yourself.