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Moving Google Authenticator: sync or manual transfer?
Do you still have the old phone, and were you using account sync? Those two questions help you choose the next step.

If you used Google-account sync, sign into that account on the new device and check your entries. If you have the old phone, you can also use the app's manual transfer process. Test the new device before clearing the old one.
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Do you have the old phone, and was sync enabled?
Can you still open the old phone, and will you keep using the same authenticator? Those questions determine the available process more directly than a general claim that one approach is safer.
A lost phone cannot create a new transfer QR. Reinstalling an app does not manufacture records that were never synchronized or backed up.
If sync was enabled, sign in on the new phone
Google documents code sync when signed into a Google account, so describing the app as local-only is inaccurate. Your own enrollment state still determines whether you used that feature.
Check entries on the destination and maintain a recovery route for the Google account itself. Avoid erasing the only device that can satisfy its second-factor requirement.
Use manual transfer if you have the old phone
Use the app's documented export and import process, checking which entries were selected. Transfer screens contain sensitive material; avoid recording or sharing them.
Verify the destination before removing the original data. Where labels repeat, compare service and account aliases rather than relying only on a matching count.
Moving to a different authenticator?
Check the destination's supported import formats first, then choose import or website re-enrollment. Keep your old device and recovery material ready, start with a familiar account, and test its login before moving on.
If easier desktop access is the goal, try the Authenticator Cloud browser extension to copy or fill codes. With Pro, sign into the same account in the app and extension and complete a sync to use the same entries on your phone.
References
See the official guides above for detailed setup instructions.
See how the Authenticator Cloud app and extensions work together for phone and computer sign-ins.

