There would not be a way to view historical data on contact moves.
If you are selecting 'add & update' when you import new contacts this will *add* membership if selecting a new group, but not remove anyone from groups.
Lastly, archiving a group can be easily confused with archiving the people inside of the group. To archive a group you would head to audience > groups > select the dropdown menu from there and select archive.
To archive the people inside of it you would just need to head inside the group and select actions > archive all.
I hope this helps provide direction here and let me know if I can help answer any questions!