I asked it before here and I was told that it may not be possible to completely replicate it on a saved search. Specifically, the function that allows you to run the aging as of a specific past date (snapshot of a past period) because the saved search takes into consideration the status of the transactions as of the time you run the saved search.
If it's even possible, then I was told it'll be very very complicated.
Anyway, the workaround is to save or send via schedule a version of the saved search results as of 12:01 of the first day of the month or whatever date you prefer to preserve a snapshot. It may not be 100% accurate if someone backdates an AR transaction in a prior period.
I hope this has changed. I'm also very interested. There's so much more that we can do if it is a saved search instead of a report.