For a BSO cube, EAS actually prompts you when you save your outline changes and asks which member in the new dimension you want the historical data sent to.
For ASO this doesn't work. One option is to reload the cube from source (if these are available) but adding in the new dimension, either in the data or via a 'header' token in the load rule(s). Another option is to export the data to a native export format, and manually edit the file to add in the name of the member in the new dimension you want to use - I do this as a single quoted token on the second line of each export file.
If you want to map historical data to different members in the new dimension this all gets more complicated.