After a successful "move" to Commons, delete the original file. Before deleting the file, use the file checksum to make sure that the file you are deleting is actually on Commons.
Only do the deletion, if the whole transfer was successful (including the template mapping)
NB: No link changes are needed to be done, if the name of the file has stayed the same
This is part of fulfilling on wish of the German-community speaking wishlist of 2013. For details see the parent task