I need to move away from Microsoft Money as it has come to the end of its life.
There seems to be lots of on line money management systems, however I need one that:
Have you looked at GnuCash? Website here
There are specific instructions for importing Microsoft Money files here: http://wiki.gnucash.org/wiki/FAQ#Q:_How_do_I_import_my_data_from_Microsoft_Money.3F
Xero can import OFX format files. They also have a free personal version, and allow automatic transaction imports with certain banks.
You also have a wide variety of export formats for your data too - along with a good API so you can write your own software to interact with the data for reporting etc.
The main online personal finance aggregation services that I know of in the UK are yodlee.com and moneydashboard.net though the latter is a new service (in Beta) powered by the former (which is more established; I've been using it on and off for around two years).
Some more info/bit of background on the two services, and some discussion in the comments, here: http://eu.techcrunch.com/2010/05/12/money-dashboard-launches-its-bid-to-become-the-mint-for-the-uk/
and this article has some screenshots/talks a bit more about what the service offers: http://thenextweb.com/apps/2010/05/14/can-money-dashboard-become-the-uks-mint-com/
I doubt they import from desktop apps like MS Money, but you may find one or other to be a viable alternative going forward..
There are also a couple of others that need you to have an account with them: Egg Money Manager (though that one saves data on a specific computer rather than being entirely web based) and First Direct have an aggregation service as well.
Been doing a bit of research into this and almost everything else I've seen is US-only unfortunately :)