The most trouble free way to move to a new machine is to use the "Set Up Assistant" on the initial boot up of the old machine:
Set up your new Mac - Apple Support
The second best way is using the "Migration Assistant" after the fact:
How to move your content to a new Mac - Apple Support
You can not pick and choose an isolated application to selective restore from the Migration Assistant.
It complicates the situation because Catalina and the new APFS may not be able to read older Time Machine backups to restore your Messages. You can try and hold the Option key and click open the TM icon in the upper right menu bar >"Browse Other Backup Disk" in an effort to restore.
If you use iCloud sync you can try signing out of icloud and back in again in an effort to repopulate the Messages
Keep all your messages in iCloud - Apple Support
Change iMessage preferences in Messages on Mac - Apple ...
Set up iCloud features on Mac - Apple Support
Restore items backed up with Time Machine on Mac -
Apple Support https://support.apple.com/guide/mac-help/restore-files-mh11422/mac
3-2-1 Backup Strategy: three copies of your data, two different methods, and one offsite.
Boot clone https://discussions.apple.com/docs/DOC-10081
How to use Time Machine to back up or restore your Mac: https://support.apple.com/en-us/HT201250
Use DiskUtility Restore feature https://support.apple.com/guide/disk-utility/restore-a-disk-dskutl14062/mac
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