General principles here
How Does a Software Update Fix Diesel Emissions? - AVID Technology
Almost certainly would involve the extent of EGR operation and injection timing [ multiple injections involved] to reduce NOx production by reducing combustion gas temperature ---an effect of which might be a slight reduction in engine thermal efficiency resulting in the loss of MPG you are experiencing.
Its doubtful if any software update customer accessible documentation would contain enough detail to satisfy your questions. Indeed it may simply consist of multiple software changes to engine ECU operating parameters downloaded directly from the factory which the dealer has no knowledge of other than which models its applicable to. All they may do is plug the car in.
Further information on recall of Mercedes-Benz vehicles with diesel engines. | Daimler
QUOTE:-
Pursuant to an administrative order by the German Federal Motor Transport Authority (KBA) --------------------------------------
According to the KBA’s interpretation of the law, certain specific calibrations of functionalities in the vehicles’ engine management must be modified.
https://www.daimler.com/documents/i...elle-diesel-rueckruf-published-by-daimler.pdf