Peter,
You can create a bacpac of your local database and store it in your local drive or on an Azure STorage Account, and then import it to Azure SQL Database using sqlpackage as explained here. That is if you want to move the whole database.
sqlpackage.exe /Action:Import /tsn:tcp:MyServer.database.windows.net,1433 /tdn:AdventureWorks /tu:AdminUser /tp:AdminPassword1 /sf:C:\temp\AW.bacpac /p:Storage=File
You can also use Data Migration Assistant to migrate/copy your local databases to Azure SQL.
In addition you can use the linked servers to insert rows on an Azure SQL Database as explained here. This method has poor performance when loading a set of rows.
You can also copy tables and rows from on premises to Azure SQL using Azure Data Factory as explained here.
ALternatively, you can use the traditional SSIS packages to send data from on-premises databases to Azure SQL as explained here.
You can also have local tables sync with Azure SQL tables using SQL Data Sync tool.
Finally you can put CSV files on Azure Storage Account and have those CSV files imported as explained here. Azure Data Factory can help you do the same.
Hope this helps.