MongoDB: move documents before capping
For the first part, expiring based on a timestamp, you will want to check out TTL collections (requires version 2.2+).  They won't cap size like a capped collection, but let's you set the constraints via time as long as you have a BSON Date type field to index on.

For the second part, automatically storing them to a new database before expiry, I think what you need is a periodic query (via cron, at, or other and in the language of your choice) to run and grab those documents before they expire.  This should just be a simple range based query over time, find the appropriate docs, insert to new DB, always at enough of an offset to avoid missing documents.  

The other approach I can think of would be to tail the oplog, look for deletes on that collection, and then transcode them into inserts on a different DB.

For extra safety you could run a secondary on a slavedelay to give an extra window if one of the approaches above failed.

I don't know of anything that will do this by default, but I believe either of the approaches above should work.