Hi,
On your computer you should have three chat.db items in ~/Library/Messages along with the Attachments folder.
In ~/Library/Containers/com.apple.iChat/Data/Library/Messages you should have the Archive folder (used if you Save Chats) and an Alias of the Attachments folders
In some cases the Containers.... item only holds two Aliases and both the Archive and Attachments folders are in the ~/Library/Messages (I have only seen two posts claiming this).
The Archive folder should be full of the Saved Chats you have made.
These are the two main sets of folders you want to Migrate to keep what Messages has.
I have not done this as I migrated an whole Mac User account from an non Intel Mac to this current iMac and this was before Messages was around.
As I have not tried it I cannot say if it displays the current iMessages and other chats in the Main Window (I suspect it will not)
Migration takes care of Ownership issues.
The other method is to manually move this items and recreate the Aliases where needed in the new locations and changing the Ownership if Required.

9:43 pm Monday; March 28, 2016
iMac 2.5Ghz i5 2011 (El Capitan)
G4/1GhzDual MDD (Leopard 10.5.8)
MacBookPro 2Gb (Snow Leopard 10.6.8)
Mac OS X (10.6.8),
Couple of iPhones and an iPad