Your MacBook Air is restarting repeatedly after the migration/restoration and returning to the same message. Since macOS is not starting normally, use macOS Recovery to check the startup disk before attempting another migration or restoration.
Disconnect the external backup drive and other non-essential devices, then shut down the Mac by holding the power button until it turns off.
If your MacBook Air has Apple silicon, press and hold the power button until Loading startup options appears, then choose Options >> Continue.
On an Intel Mac, turn it on while holding Command (⌘)-R. Apple: How to start up from macOS Recovery
In Recovery, open Disk Utility, choose View >> Show All Devices, and run First Aid on each applicable volume, followed by the container and physical storage device.
If First Aid reports that the disk is OK or repairs it successfully, restart the Mac. Apple: Repair a storage device in Disk Utility
If the restart loop continues, return to Recovery and choose Reinstall macOS.
Apple indicates that reinstalling macOS from Recovery can be done while retaining personal data, apps, and user settings. Apple: How to reinstall macOS
If First Aid cannot repair the disk, or the internal startup disk does not appear in Disk Utility, do not proceed with another restoration yet.
Apple indicates that an unrepairable disk may require erasing and reinstalling macOS, while a disk that does not appear or cannot be erased may require service. Apple: Repair a storage device in Disk Utility
Once macOS starts normally, if the external drive contains a Time Machine backup, Migration Assistant can be used to transfer the backed-up information.
Apple recommends reinstalling macOS first when necessary and then using Migration Assistant to restore information from a Time Machine backup. Apple: Restore your Mac from a backup
The exact next step therefore depends on whether the external drive contains a Time Machine backup and whether the MacBook Air has Apple silicon or an Intel processor.
