originaldating - Updating maximum flow
I do not know which max flow algorithm you are using so I can not be more specific, but it is possible that after adding the new edge you violate the property of the algorithm and thus you find the max flow in a sub-optimal way.
Still you've already processed most of the graph so this should not be a too big problem.
Then run the algorithm you use to find augmenting path until you find non-zero flow and that's it.
Most of the max flow will already have been found so this should(in theory) not be too slow.
Microsoft Flow includes commonly used connections, including Share Point, SQL Server, Office 365, One Drive for Business, Salesforce, Excel, Dropbox, Twitter, and more.