Welcome to Microsoft Q&A! Thanks for posting the question.
As per the error message "There is an in progress azure migration for site 'xxx'. Please try again later" it looks like there is already a customer initiated or backend operation that is currently going on and self-resolved by itself once the upgrades are completed. The 409 conflict is expected behaviour when there is already on going operation and you tried to perform another operation while the previous operation is not completed.
If you have observed this issue for a long time then please share the CorrelationRequestId for the operation that you are performing along with the timeframe in UTC so I can check the backend logs and if the issue is due to the backend resource then I can engage my team to mitigate the issue.