Migrating files from new computer running is Sonoma 14.2.1 to MacBook Air 2019 running macOS 10.13 High sierra

Migration is not going to work the way you tried it. Generally you cannot migrate 'backwards' meaning from a recent version of macOS to an earlier version. Even Target Disk Mode (TDM) can be difficult because earlier versions of macOS don't recognize the newer APFS disk format.

If you just want to transfer files (not apps or users), I suggest trying this -

On both Macs, enable file sharing and make sure SMB is enabled. (On the MBAir running High Sierra, see System Preferences > Sharing > Options > Share files and folders using SMB. There is a similar setting in Sonoma.)

Then from the MBAir, log into the Sonoma iMac, select the drive or folder you want to copy from and copy the files you want from the Sonoma Mac to your MBAir.

This dog won't hunt on many different levels, some of which have already been explained. Don't go there.

MacOS migrates Mail installations forward, not backward, so the Mail application on High Sierra may not work. Apple may entirely redesign .plists (preferences files) for a Sonoma installation and copying these back to High Sierra would overwrite the old files with content not expected on High Sierra. Things will inexplicably (to you) just break and be impossible for anyone here to help you fix.

You would be moving from a high security operating system to a comparatively much lower security platform where many security protocols have been substantially deprecated by existing web sites, mail servers, financial institutions, and many application developers have simply abandoned High Sierra as an unsupportable platform.

Running MA on both ends isn't likely to working that far backwards, if anything will work with that spread of OS versions, which I doubt/// it'd be putting the newer one in Target Mode, but the Disk structures are quite different.

Depending what you want to transfer, iCloud Drive & iCloud Keychain may be of some help.