Copy Chatter feed from one object to another (of other SObject Type)

I need to migrate data from one object type to another. Including their Chatter feeds. First task is simple, but the second?

How would I best approach this? Are there AppExchnage tools for that or existing Apex code? What would I have to think of?

1. Can I preserve the posting dates?
2. Can I just relink post/comment attachments?
3. How to cope with @mentions and Likes?
4. What will such a migration do to reputation scores?

Answers:

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Method 1

I ended up building my own tool which can be found on Github: https://github.com/Up2Go/CopyChatterFeed

All methods was sourced from stackoverflow.com or stackexchange.com, is licensed under cc by-sa 2.5, cc by-sa 3.0 and cc by-sa 4.0